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Canard-like explosion of limit cycles in two-dimensional
piecewise-linear models of FitzHugh-Nagumo type
.
NJIT CAMS Technical Report
1112-4
Horacio G. Rotstein 1∗, Stephen Coombes 2, Ana Maria Gheorghe 2,
1 Department of Mathematical Sciences
New Jersey Institute of Technology
Newark, NJ 07102, USA2 School of Mathematical Sciences
University of Nottingham
Nottingham NG7 2RD, UK
Abstract
We investigate the mechanism of abrupt transition between small and large amplitude oscillations
in fast-slow piecewise-linear (PWL) models of FitzHugh-Nagumo (FHN) type. In the context of
neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials
(spikes) respectively. The minimal model that shows such phenomenon has a cubic-like nullcline (for
the fast equation) with two or more linear pieces in the middle branch and one piece in the left and
right branches. Simpler models with only one linear piece in the middle branch or a discontinuity
between the left and right branches (McKean model) show a single oscillatory mode. As the number
of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the
minimal model, we investigate the bifurcation structure, we describe the mechanism that leads to
the abrupt, canard-like transition between subthreshold oscillations and spikes, and we provide an
∗Also, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology. E-mail: horacio@njit.edu
1
analytical way of predicting the amplitude regime of a given limit cycle trajectory which includes the
approximation of the canard critical control parameter. We extend our results to models with a larger
number of linear pieces. Our results for PWL-FHN type models are consistent with similar results for
smooth FHN type models. In addition, we develop tools that are amenable for the investigation of a
variety of related, and more complex, problems including forced, stochastic and coupled oscillators.
1 Introduction
In a two-dimensional relaxation oscillator, the canard phenomenon (or canard explosion) refers to
the abrupt increase in the amplitude of the limit cycle created in a Hopf bifurcation as a control
parameter crosses a very small critical range which is exponentially small in the parameter defining
the slow time scale [1, 2, 3, 4, 5, 6, 7]. Depending on whether the Hopf bifurcation is supercritical
or subcritical, the small amplitude limit cycles are either stable or unstable respectively. The large
amplitude, relaxation-type limit cycles are always stable.
We illustrate the canard phenomenon for both cases in Fig. 1 for a smooth, fast-slow system of
FitzHugh-Nagumo (FHN) type of the form
v′ = f(v)− w,
w′ = ǫ [α v − λ− w ],(1)
where α, λ and ǫ are constants, 0 < ǫ ≪ 1, α > 0, and f(v) is a cubic-like function. We used ǫ = 0.01
and the prototypical cubic function f(v) = −h v3 + a v2 with h = 2 and a = 3 whose maximum and
minimum occur at (0, 0) and (1, 1) respectively. This canonical choice ensures that the large amplitude
oscillations are O(1). The parameters α and λ control the slope of the w-nullcline Nw = α v − λ and
its position relative to the v-nullcline Nv = f(v) respectively. In the context of neuroscience, the
variables v and w correspond to a dimensionless voltage and a recovery variable respectively, and the
parameter λ can be thought of as a dimensionless externally applied (DC) current, after redefining
w → w + λ. In Appendix A we provide a technical discussion about the Hopf bifurcation and canard
phenomenon for these systems.
For λ = 0 the v- and w-nullclines in Fig. 1 (Nv and Nw respectively) intersect at the minimum of
the cubic nullcline Nv. For the parameters we used, the Hopf bifurcation is supercritical (subcritical)
for α > 3 (α < 3) (see Appendix A). Consequently, the Hopf bifurcation is supercritical in Figs. 1-A
(α = 4) and subcritical in Figs. 1-B (α = 2). The left panels show sketches of the graphs of the limit
cycle amplitude versus the control parameter λ (A = 0 indicates a fixed point). In both cases, the
Hopf bifurcation occurs as λ crosses λH = O(ǫ) > 0 (see Appendix A). For values of λ < λH (λ > λH)
the fixed-points are stable (unstable). In the supercritical case (Fig. 1-A), the fixed-point is the only
attractor for λ < λH and the system is excitable [8, 9, 10, 11]. In the subcritical case (Fig. 1-B), there
is bistability for a range of values of λ where large amplitude oscillations and a fixed point coexist.
The small amplitude limit cycles created at λ = λH are stable in the supercritical case (Fig.
1-A) and unstable in the subcritical case (Fig. 1-B). Their amplitude increases slowly for values of
λ close to λH . In the supercritical case (Fig. 1-A) this happens as λ increases. As λ crosses from
2
A
λ
A
supercritical
stableunstable
0λ
Hλ
c
−1 −0.5 0 0.5 1 1.5 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
v
w
α = 4 λ = 0.0078
N
v(v)
Nw
(v)
Tr
−1 −0.5 0 0.5 1 1.5 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
v
w
α = 4 λ = 0.0079
N
v(v)
Nw
(v)
Tr
B
λ
A
subcritical
stableunstable
0λ
cλ
H
−1 −0.5 0 0.5 1 1.5 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
v
w
α = 2 λ = 0.0027
N
v(v)
Nw
(v)
Tr
−1 −0.5 0 0.5 1 1.5 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
v
w
α = 2 λ = 0.0028
N
v(v)
Nw
(v)
Tr
Figure 1: Supercritial and subcritical canard phenomenon (canard explosion) for the FHN
model (1) with f(v) = −2 v3 + 3 v2, ǫ = 0.01 and various values of α and λ. Left panels: bifurcation
diagrams. A = 0 corresponds to fixed-points, A > 0 corresponds to the amplitude of limit cycles, λH
and λc indicate the Hopf bifurcation and canard critical points respectively. Middle and right panels:
phase-planes for representative values of λ. Nv and Nw represent the v- and w-nullclines respectively and
Tr represents the trajectory. A Supercritical case: λH ∼ 0.0025 and λc ∼ 0.0078. Small amplitude limit
cycles are stable. B Subcritical case: λH ∼ 0.0035 and λc ∼ 0.00278. Small amplitude limit cycles are
unstable. The trajectory shown in the middle panel displays damped oscillations and converges to the
stable fixed-point.
3
λ = 0.0078 to λ = 0.0079 (middle to right panels), the limit cycle “explodes”. The small (exponentially
small in ǫ) range of values of the control parameter λ over which this canard explosion occurs can
be approximated by the canard critical value λc = 0.00782. In the subcritical case (Fig. 1-B), the
unstable small amplitude limit cycle (not shown) explodes as λ decreases below λc = 0.00278. The
values of λc were computed using the formulas given in Appendix A. One feature of the canard
phenomenon is that trajectories evolve in close vicinities of the unstable (middle) branch of the v-
nullcline for a significant amount of time before moving either to the left or to the right to generate
small and large amplitude oscillations respectively (see Fig. 1-A for example).
The canard phenomenon for smooth two-dimensional systems has been investigated by various
authors. It was discovered by Benoit et al. [4] for the Van der Pol (VDP) oscillator where the w-
nullcline is a vertical line (see Appendix A.3). They used non-standard analysis techniques. Eckhaus
[2] and Baer et al. [3] investigated the canard phenomenon using asymptotic techniques. In particular
they found expressions for λc for VDP- and FHN-type equations. The canard phenomenon has also
been studied by Dumortier and Roussarie [1] and by Krupa and Szmolyan [5, 6] for more general
two-dimensional slow-fast systems. These results assumed a set of non-degeneracy conditions which
in particular imply that the v-nullcline has to be locally parabolic at its minimum. For future use, we
sketch some of these results in Appendix A.
The goal of this paper is to investigate the mechanism leading to the canard explosion in two-
dimensional, piecewise linear (PWL) systems of the form (1) with f(v) substituted by a cubic-like
PWL function. PWL caricatures of nonlinear models have been fruitfully used in a number of different
branches of the applied sciences ranging from biology to mechanics as a way to provide new insights
into the dynamics of smooth models or as a convenient way to explicitely analyze them when a general
set of mathematical techniques is not available. For example in neuroscience the McKean model [12]
may be regarded as a variant of the FitzHugh-Nagumo model [13] that provides a planar model of
an excitable cell in which the dynamics is broken into simpler linear pieces. This has allowed for a
number of results about the existence and stability of periodic orbits and traveling waves to be obtained
[14, 15, 16, 17, 18, 19] and insight into action potential generation and response to stimulation to be
obtained [20]. An extension of this approach to other single neuron models, including the Morris-Lecar
model, has recently been pursued by Tonnelier and Gerstner [21] and Coombes [22]. In a mechanical
setting PWL modeling has helped to shed light on the motion of rocking blocks [23] and models of
suspension bridges [24], and indeed impacting systems in general (see the recent book by di Bernardo
et al. [25]). Many of the techniques for analysing mechanical systems have been taken over to the
study of oscillatory electronic circuits and in particular the analysis of non-smooth bifurcation such as
those of border-collision type [26]. In the systems biology context techniques originally developed by
Filippov [27] (for ordinary differential equations with a discontinuous vector field) have been applied
to gene regulatory networks with switch-like interactions (arising in the limit of a steep sigmoidal
nonlinearity) [28, 29]. From a more mathematical perspective PWL systems have allowed for the
analysis of limit cycles [30] and their bifurcations [31, 32] and have been shown numerically to support
canard solutions [33].
The investigation of the dynamics of PWL oscillators, and in particular the abrupt transition
between limit cycle amplitude regimes provides complementary information to previous studies on
4
analogous smooth systems. These studies have focused on the existence of the so called canard
solutions (including the so called maximal canards which exist for values of the control parameter
in the small transition range between small and large amplitude limit cycles) and the conditions
under which canard explosions occur, and less attention has been paid to the actual evolution of
trajectories for values of λ near the critical value for the abrupt transition (λc). Understanding this
is important not only for the understanding of the dynamics of single oscillators but also for the
understanding of the effects that external inputs (e.g., pulsatile, sinusoidal, noisy and synaptic-like)
exert on single oscillators and the dynamics of oscillatory networks. For small enough values of ǫ,
trajectories corresponding to single oscillators can be approximated over most of their period either
by the stable branches of the v-nullclines, which approximate the slow manifolds, or by the horizontal
fast fibers. These phases of the oscillation are typically robust to external perturbations.
By contrast, perturbations in a range of phases in which the trajectory is evolving in a close
vicinity of the unstable branch of the v-nullcline can lead to disparate effects where the resulting
amplitude regime (small or large) of the perturbed system is different from that of the unperturbed
one. More specifically, using neuroscience terminology, spiking can be created or suppressed, or delayed
/ advanced by a significant amount of time with respect to the unperturbed oscillation. The canard
critical value λc (see Appendix A) provides information on whether the limit cycle trajectory of an
unperturbed oscillator crosses the unstable branch of the v-nullcline (small amplitude oscillations)
or not (large amplitude oscillations). However, when the system is perturbed the relative position
between nullclines changes and so do the fixed-point and the canard-critical value. For time-dependent
perturbations, the effects of these changes in evolving trajectories are difficult to predict as are the
resulting amplitude regimes of perturbed trajectories. In particular, the canard critical value, that
becomes time-dependent, is no longer an appropriate tool for this purpose. Asymptotic techniques can
be used to provide a description of trajectories for small values of ǫ. However, even for single oscillators
[2, 3] their description becomes complicated and provides little intuition on their evolution. In addition,
as the value of ǫ increases to intermediate values still smaller than O(1) these approximations become
less accurate.
In Section 2, we introduce some notation on PWL systems of FHN type, where the function f(v)
in (1) is substituted by a cubic-like PWL function, and we overview, for future use, the solutions to
the linear regimes corresponding to each one of the linear pieces in a PWL system. In Section 3, we
investigate the dynamics of PWL systems having two linear pieces in the middle branch of the cubic-
like function and one linear piece in the other two. We refer to them as PWL1,2,1 systems. These are
the simplest PWL models for which canard explosions occur. In linear systems with only one linear
piece in the middle branch, limit cycles have a single (large) amplitude regime. We first present the
bifurcation structure of PWL1,2,1 systems. Then, we investigate the mechanism leading to the canard-
like explosion of limit cycles. Although solutions for PWL systems can be computed analytically, the
insight they provide into the underlying dynamics is poor. We use dynamical systems tools to explain
these dynamics and answer various relevant questions about the abrupt transition between limit cycle
amplitude regimes. In addition, we provide a way of calculating analytical approximations to the
canard critical value λc and discuss the accuracy of these approximations and the dependence of λc
on other relevant model parameters. Finally, we show that, although the amplitude of the small limit
5
cycle changes with the model parameters, the period is independent of λ. In Section 4, we extend our
results to PWL systems having three linear pieces in the middle branch and one linear piece in the
other two. We discuss the implications of our results for more general and complex models in Section
5.
2 Piecewise linear models of FitzHugh-Nagumo type
We consider the following piecewise linear (PWL) models of FitzHugh-Nagumo (FHN) type where the
cubic-like smooth function f(v) in system (1) is substituted by a PWL caricature fpwl(v):
v′ = fpwl(v)− w,
w′ = ǫ [α v − λ− w ].(2)
We use prototypical PWL functions fpwl(v) with minimum (vmin, wmin) = (0, 0) and maximum
(vmax, wmax) = (1, 1). As in the smooth case discussed above, this choice ensures that large amplitude
oscillations are O(1). Each branch of fpwl(v) consists of one or more linear pieces Lj indexed by j.
The v- and w-nullclines of model (2) are given by
Nv = fpwl(v) and Nv = α v − λ (3)
respectively.
2.1 Construction of piecewise linear functions
We refer to a cubic-like PWL function having M , Ml and Mr linear pieces in the middle, left and
right branches respectively as a PWLMl,M,Mr function. If the number of linear pieces in all branches
of f(v) is equal (Ml = Mr = M), then we refer to this function as PWLM . We use an analogous
terminology for the corresponding models. We illustrate this in Fig. 2 for a PWL1,2,1 model
In the process of building the PWL function fpwl(v) we use the following notation:
• We partition the interval [vmin, vmax] into M segments with end-points vj , for j = 0, . . . ,M ,
where v0 = vmin, vM = vmax and vj−1 < vj .
• We call Lj (j = 1, . . . ,M) the linear piece that has endpoints (vj−1, wj−1) and (vj , wj).
• We call ηj, j = 1, . . . ,M the slope of the linear piece Lj.
• Given the values v0, . . . , vM and the first M − 1 slopes (η1, . . . , ηM−1) we calculate wj = ηj (vj −vj−1) + wj−1 for j = 1, . . . ,M − 1. The slope of the last piece in the middle branch is given by
ηM = (wM − wM−1)/(vM − vM−1).
• We proceed in a similar manner with the left and right branches. In the left branch j =
−Ml − 1, . . . , 0 and in the right branch, j = M, . . . ,M +Mr. Alternatively, we use the notation
Ll,j and Lr,j as illustrated in Fig. 2.
6
2.2 PWL approximations of smooth vector fields
In order to link the cubic-like PWL functions fpwl(v) in eq. (2) to the corresponding smooth functions
f(v) in eq. (1) we use M = Ml = Mr and proceed as follows:
• We call ∆v = (vmax − vmin)/M and vj = j∆v for j = −Ml, . . . ,M +Mr with M = Ml = Mr.
• We call wj = f(vj) for j = −Ml, . . . ,Mr.
• We call ηj = (wj − wj−1)/(vj − vj−1) for j = −Ml + 1, . . . ,M +Mr.
As M increases, fpwl(v) approaches the corresponding smooth function f(v). Fig. 3 illustrates the
approximation of the solutions to the corresponding PWLM systems (2) to the smooth system (1) as
M increases.
In both Figs. 3-A and -B, the top-left panels show the phase-plane diagrams corresponding to
the smooth vector fields for two different values of λ which are close to the canard critical point λc.
The parameters α and ǫ are as in Fig. 1-A (supercritical case). In Fig. 3-A, λ > λc and the smooth
system exhibit large amplitude oscillations. In Fig. 3-B, λ < λc and the smooth system exhibits
small amplitude oscillations The remaining panels in both Figs. 3-A and -B correspond to the PWL
approximations for a number of linear pieces that increases from the top-middle to the bottom-right
panels.
As M increases, the dynamics of the PWL system approach the dynamics of the corresponding
smooth systems. For λ > λc (Fig. 3-A), the limit cycle is in the large amplitude regime for all values
of M . In contrast, for λ < λc (Fig. 3-B), the limit cycle is in the large amplitude regime for small
values of M ( ≤ 9) and the transition to the small amplitude oscillations regime occurs for M = 10.
We explain the corresponding mechanism in the following sections.
2.3 Dynamics of the basic linear components
In order to investigate the dynamics of a PWL system such as (2) it is convenient to consider it divided
into a number of linear regimes Rj associated to the linear pieces Lj . A linear regime Rj is a strip in
the phase-plane bounded by the vertical lines crossing the end-points of the linear piece Lj , v = vj−1
and v = vj respectively (see Fig. 2). More specifically, a point (v,w) in phase-plane belongs to Rj if
vj−1 ≤ v ≤ vj where vj−1 and vj are the v-coordinates of the left and right endpoints of the linear
piece Lj . Note that two regimes corresponding to linear pieces with a joint endpoint interesect at the
vertical line containing this point.
The dynamics of each linear regime Rj are governed by a linear system of the form
v′ = ηj (v − vj−1) + wj−1 − w,
w′ = ǫ [α v − λ− w],(4)
where (vj−1, wj−1) and ηj are the left end-point and slope of the corresponding linear piece Lj respec-
tively. The initial conditions v0,j and w0,j in each regime are equal to the values of v and w at the
end of the previous one. More specifically, if Ri is the regime prior to Rj (either Rj−1 or Rj+1), and
7
the solution in Ri arrives at vi (v-coordinate of the joint point between Li and Lj) at a time ti, then
the initial conditions for system (4) corresponding to the linear regime Rj are v = vi and w = w(ti).
The solutions to (4) can be calculated using standard methods [34]. We present them below and
illustrate them in Section 3 tied to our explanation of the canard explosion. For simplicity, we drop
the indices from the endpoints (v, w), the slope η and the initial conditions (v0, w0). The fixed point
for system (4) is given by
v =λ− η v + w
α− ηw =
λ η − α η v + α w
α− η. (5)
The eigenvalues are given by
r1,2 =η − ǫ ±
√
(η + ǫ)2 − 4 ǫ α
2. (6)
The fixed-point (v, w) is stable (unstable) for η < ǫ (η > ǫ). There are two critical slopes given by
η+cr = −ǫ+ 2√ǫ α and η−cr = −ǫ− 2
√ǫ α (7)
such that the eigenvalues are complex for η ∈ ( η+cr, η−cr ) and real otherwise; i.e., the fixed point (v, w)
is a focus for η ∈ (η−cr, η+cr) and a node otherwise. We show the corresponding stability diagram for
ǫ = 0.01 in Fig. 4-A. For a fixed value of α, as the slope of the linear piece η increases, the fixed
point (v, w) changes from a stable node through stable and unstable spirals, to an unstable node. The
horizontal line η = ǫ corresponds to the Hopf bifurcation points. As ǫ increases (“from red to blue”
in Fig. 4-B) the Hopf bifurcation line moves upwards and the interval of values of η (range of slopes
of the linear pieces) corresponding to the spiraling behavior widens.
The solution to system (4) for real eigenvalues r1 and r2 (( η + ǫ )2 − 4 ǫ α > 0) is given by
[
v
w
]
= c1
[
1
r2 + ǫ
]
er1t + c2
[
1
r1 + ǫ
]
er2t +
[
v
w
]
(8)
with
c1 =(v0 − v) (r1 + ǫ)− (w0 − w)
r1 − r2and c2 =
−(v0 − v) (r2 + ǫ) + (w0 − w)
r1 − r2. (9)
In Fig. 4-C we show graphs of the real roots (r1 and r2) as a function of the slope of the linear
pieces η for various values of α and ǫ = 0.01. For η < 0 (left and right branches of the cubic-like
PWL function) the corresponding fixed-point is stable (both eigenvalues are negative) while for η > 0
(middle branch of the cubic-like PWL function) the fixed-point is unstable (r2 may become negative
as η increases for small enough values of α). For small values of ǫ and large enough values of η there
is a time scale separation (fast-slow system) which dissappears as η decreases or α increases.
The solution to system (5) for complex eigenvalues r1 and r2 (( η + ǫ )2 − 4 ǫ α < 0) is given by
[
v
w
]
= c1
[(
1
β
)
cosµ t +
(
0
µ
)
sinµ t
]
e r t +
8
+ c2
[(
1
β
)
sinµ t −(
0
µ
)
cosµ t
]
e r t +
[
v
w
]
(10)
with
c1 = v0 − v, c2 =(v0 − v) (η + ǫ)− 2 (w0 − w)
2µ(11)
where
µ =√
4α ǫ− (η + ǫ)2 / 2. (12)
and
r =η − ǫ
2, β =
η + ǫ
2. (13)
In Fig. 4-D we show graphs of the natural frequency µ as a function of the slope of the linear piece
for various values of α and ǫ = 0.01.
3 The dynamics of PWL1,2,1 models of FHN-type
Here we investigate the mechanisms of generation of both small and large amplitude limit cycles and
the abrupt transition between them (canard-like explosion) in PWL1,2,1 models of the form (2) where
the cubic-like function fpwl(v) is as illustrated in Fig. 2. We show an example in Fig. 5 (panels A
and B).
The middle branch of fpwl(v) has two linear pieces, L1 and L2, with slopes η1 and η2 respectively.
We considered various representative values of the lengths and slopes of these linear pieces. The
maximum and minimum of fpwl(v) are located at (0, 0) and (1, 1) respectively. As mentioned above,
this canonical choice ensures that large amplitude oscillations are O(1). The left and right branches
have one linear piece each, Ll,1 and Lr,1, with slopes ηl,1 and ηr,1 respectively. We used ηl,1 = ηr,1 = −1.
As we will see, the number of pieces and the choice of slopes in both the left and right branches have
little effect on the canard-like explosion for small enough values of ǫ.
In a PWL1,2,1 model, the two linear pieces in the middle branch (L1 and L2) join at the point
(v1, w1). For the example shown in Fig. 2, (v1, w1) = (0.3, 0.09). From eqs. (5), if the intersection
between the two nullclines, Nw(v) = α v − λ and Nv(v) = fpwl(v), occurs on the linear piece L1, then
the fixed point (v, w) is given by
v =λ
α− η1and w =
λ η1α− η1
(14)
where η, v and w have been substituted by η1, v0 = 0 and w0 = 0 in eqs. (5).
For λ = 0, (v, w) = (0, 0). As λ increases (decreases), the w-nullcline Nw(v) = α v − λ moves to
the right (left), and so does the fixed point (v, w). This remains true if the intersection between the
two nullclines occurs on the linear piece L2. In this case, eq. (14) has to be modified accordingly.
9
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
Nv(v)
Nw
(v)
Ll,1
Lr,1
Ll,1
L2
Rr,1
R2
R1R
l,1
Figure 2: Nullclines (Nv and Nw), linear pieces and linear regimes for a PWL1,2,1 cubic-like model (2) with
α = 4 and λ = 0. The slope of the linear pieces in Nv are (from left to right): ηl,1 = −1 (left branch),
η1 = 0.3 and η2 = 1.3 (middle branch), and ηr,1 = −1 (right branch). The linear pieces in the middle
branch join at (v1, w1) = (0.3, 0.09).
(v, w) η r1 r2 r µ set
Ll,1 (0, 0) -1.0 -0.052 -0.958 S - N I
Lr,1 -1.0 -0.052 -0.958 S - N I
L1 (0.3, 0.09) 0.3 0.145 0.126 U - F II
L2 (1, 1) 1.3 1.269 0.021 U - N II
L1 (0.4, 0.12) 0.3 0.145 0.126 U - F III
L2 (1, 1) 1.467 1.439 0.018 U - N III
L1 (0.9, 0.27) 0.3 0.145 0.126 U - F IV
L2 (1, 1) 7.3 7.295 -0.045 U - N IV
Table 1: PWL1,2,1 models of FHN type for α = 4 and ǫ = 0.01. The parameters correspond to three models
(sets I/II, sets I/III and sets I/IV) with the same left and right branches with a single linear piece each
(set I). For each linear piece Lj , the table shows the right endpoints (v, w), the slope η, the eigenvalues of
the corresponding linear regime (real eigenvalues r1 and r2, or real and imaginary parts, r = (η− ǫ)/2 and
µ respectively if the eigenvalues are complex). The transition from unstable foci (U-F) to unstable nodes
(U-N) occurs at η+cr = 0.39. For the parameters we used, the transition from stable foci (S-F) to stable
nodes (S-N) occur at η−cr = −0.41.
10
A
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 1 α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 3 α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 10 α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 100 α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 1000 α = 4 λ = 0.008
N
v(v)
Nw
(v)
Tr
B
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 1 α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
wM = 3 α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 9 α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 10 α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
−0.5 0 0.5 1 1.5
0
0.5
1
1.5
v
w
M = 100 α = 4 λ = 0.007
N
v(v)
Nw
(v)
Tr
Figure 3: Solutions to PWLM models approximate the corresponding smooth FHN model with
f(v) = −2 v3 + 3 v2 as the number of linear pieces M increases. A: λ = 0.008. B: λ = 0.007. The
top-left panels in A and B show the phase-planes for the smooth FHN model in the relaxation oscillations
and small amplitude oscillations regimes respectively. The corresponding values of λ are close to the canard
critical point λc ∼ 0.0078. As the number of linear pieces M increases (from the top-middle panel to the
bottom-right panel) in both A and B, the solutions to the PWLM models approximate the solution to the
smooth FHN model (shown in the top-left panel).11
A B
0 5 10 15−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
α
η
ε = 0.01
stable spirals
stable nodes
unstable spirals
unstable nodes(η + ε)2 − 4 ε α = 0
η − ε = 0
0 5 10 15−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
α
η
ε = 0.01ε = 0.05ε = 0.1
C D
−3 −2 −1 0 1 2 3−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
η
ε = 0.01
r1
r2α = 1
α = 4
α = 10
−1 −0.5 0 0.5 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
η
µ
ε = 0.01
α = 1α = 4α=10
Figure 4: Stability diagrams and eigenvalues for the basic linear components in a PWL model
of FHN type. A and B. Stability diagrams. For a fixed value of α, as η increases the fixed-point (v, w)
changes from a stable node, through a stable and unstable focus, to an unstable node. C. Real eigenvalues
(r1 and r2). They correspond to values of α and η satisfying (η + ǫ)2 − 4α ǫ > 0. D. Natural frequencies
µ for complex eigenvalues. They correspond to values of α and η satisfying (η + ǫ)2 − 4α ǫ < 0.
12
A B
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.029
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.03
Nv(v)
Nw
(v)
C
−1 −0.5 0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4
N
v(v)
Nw
(v)
λ=0.02931λ=0.029315L
i
0.2 0.25 0.3 0.35 0.4 0.450.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
v
w
α = 4
Nv(v)
Nw
(v)
λ=0.02931λ=0.029315L
i
Figure 5: Canard phenomenon in a PWL1,2,1 model of FHN type. A. Small amplitude limit
cycle for λ = 0.029. B. Large amplitude limit cycle for λ = 0.03. C. Phase-plane diagram showing the
inflection line L1 (or w = w2(v)) separating between a small amplitude limit cycle (λ = 0.02931) and a
large amplitude limit cycle (λ = 0.029315). The inflection line is defined for values of v ∈ [0.3, 1]. For
clarity, we extended it beyond this domain. The right panel is a magnification of the left one. We used
(v1, w1) = (0.3, 0.09), η1 = 0.3, ǫ = 0.01 and α = 4. (See also Table 1, sets I and II).
13
For each linear piece, the eigenvalues are given by (6) with η substituted by the corresponding
value of ηj. Eigenvalues and eigenvectors depend on the linear piece on which the fixed-point (v, w)
is located but not on the precise location of the fixed-point on that linear piece; i.e., as λ increases,
the eigenvalues and eigenvectors remain unchanged as long as the fixed point remains on the same
linear piece. For future use, we present in Table 1 (sets I and II) the eigenvalues (r1 and r2), or their
real and imaginary parts (r and µ) if they are complex, for the fixed-points in Fig. 2. Fixed-points
located on the left and right branches are stable nodes (r1 < 0 and r2 < 0), fixed-points located on
the linear piece L1 are unstable foci (r > 0 and µ 6= 0), and fixed-points located on the linear piece
L2 are unstable nodes (r1 > 0 and r2 > 0).
3.1 Assumptions and properties of the linear regimes
In PWL1,2,1 systems, the linear regimes R1 and R2 satisfy some geometric and dynamic constraints.
First, by construction, the slopes of L1 and L2 (linear pieces in the middle branch) and the v-coordinate
v1 of the point joining them are related by
η2 =1− η1 v11− v1
. (15)
The exact form of this equation depends on our canonical choice of the maximum and minimum of
fpwl(v). Similar expressions can be found for other values of (vmin, wmin) and (vmax, wmax) (see Section
2.1). Secondly, from (15), if η1 < 1, then η2 > 1. This also follows from geometric considerations. The
slope of the line L joining the minimum (0, 0) and maximum (1, 1) of the cubic like PWL1,2,1 function
is η = 1. (This line would be the middle branch of a cubic-like PWL1,1,1 function as the ones shown
in Figs. 3-A and -B (top-left panels)). The two linear pieces L1 and L2 in Fig. 2 can be thought of as
resulting from the line L “breaking” into two linear pieces with slopes smaller and larger than 1 (the
slope of L) respectively.
Finally, the two regimes R1 and R2 have fixed-points with different stability properties. More
specifically, if R1 has an unstable focus (0 < η1 < η+cr with η+cr ≤ 1), then R2 has an unstable node
(η2 > η+cr), and viceversa. (The critical slope η+cr is given by the first equation in (7)). This is true
since, by assuming the contrary (η2 ≤ η+cr), substituting in (15) and rearranging terms we arrive to
η1 ≥ η+cr, contradicting our previous assumption. The converse is also true and follows from similar
arguments by noting that eq. (15) holds if η1and η2 are exchanged.
We show in Section 3.6 that PWL1,2,1 systems do not display small amplitude oscillations if R1 has
an unstable node; i.e., a necessary (but not sufficient) condition for the occurrence of small amplitude
oscillations is that R1 has a focus. Since the occurrence of the canard phenomenon requires the
existence of small amplitude oscillations, here we consider parameters α, ǫ and η1 satisfying η1 < η+cr.
We also require that η1 < α in order to allow for the intersection between the linear piece L1 and the
w-nullcline (Nw(v) = αv − λ) to occur for appropriate values of λ. In addition, we assume that the
system has only one fixed-point.
Note that these assumptions are analogous to the requirement that the smooth system (1) under-
goes a supercritical Hopf bifurcation. Consider the cubic function f(v) = −hv3 + av2 used in Fig.
14
1 (h = 3 and a = 2). As shown in Appendix A.4, for fixed values of h and a, there exists a critical
value αcrit = 2a2(3h)−1 such that the Hopf bifurcation is supercritical for α > αcrit and subcritical
for α < αcrit. As α decreases below αcrit, the smooth system loses its ability to generate stable small
amplitude limit cycles. (For the choice of parameters in Fig. 1, αcrit = 3). For a PWL1,2,1 and a fixed
value of η1, since η+cr is a decreasing function of α, a large enough decrease in the value of α causes
η+cr to decrease below η1 thus changing the stability properties of the corresponding fixed-point of the
linear regime R1 from an unstable focus to an unstable node and hence small amplitude oscillations
are no longer possible (see Section 3.6).
3.2 Bifurcation structure
In Fig. 6-A we present the limit cycle amplitude diagram for the PWL1,2,1 system corresponding to
Fig. 2 and ǫ = 0.01 as a function of the control parameter λ. Linear stability properties of fixed-points
were determined as in section 2.3. Periodic solutions are constructed using the explicit formulas for
trajectories given by the solutions in Section 2.3, matching continuously across sections where v = vj ,
and enforcing periodicity of solutions. These constraints generate a set of nonlinear algebraic equations
which we solve numerically as in the earlier work by Coombes [22] to determine the period of solution,
T , and the maximum and minimum values of the orbit.
For λ < 0, there is a stable node lying on the left branch of the v-nullcline (linear piece Ll,1). As
λ increases, the fixed-point (14) moves to the right and crosses the minimum of the v-nullcline when
λ = 0. For 0 < λ < 0.029 the system has small amplitude limit cycles qualitatively similar to the ones
shown in Fig. 5-A. The amplitudes of these limit cycles increase with λ. For λ = 0.03 the system
has a large amplitude, relaxation type limit cycle (Fig. 5-B). A major difference between these small
and large amplitude limit cycles is that trajectories cross the middle branch of the v-nullcline in the
former while they don’t, and move away from R1, in the latter. This is analogous to the smooth case.
With increasing values of ǫ it is possible for the branch of small amplitude oscillations to develop
a fold so that stable small and large amplitude oscillations can coexist (and preclude the canard
phenomenon). We illustrate this in Fig. 6-B. This phenomenon is not observed in smooth systems
for the same parameter values (α = 4 and ǫ = 0.1). A similar folding in the PWL system occurs for
other parameter sets as we illustrate in Fig. 7-A for α = 2 and ǫ = 0.1. However, differently from the
α = 4 case, for the α = 2 case, the folding phenomenon persists in the corresponding smooth system
Fig. 7-B.
The period of a small amplitude limit cycle is independent of λ
Here we show that for a periodic orbit like that shown in Fig. 5-A which visits only two distinct
regimes R1 and R2 in the middle branch the period of the orbit is independent of λ.
The piecewise dynamical system (4) may be written in the form
z′j = Ajzj + bj, zj =
[
v
w
]
, (v,w) ∈ Rj , (16)
15
A
−1 0 1 2 3 4−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
λ
v
α = 4 ε=0.01
stable fixed−pointunstable fixed−pointstable limit cycle
−0.02 0 0.02 0.04 0.06 0.08 0.1−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
λ
v
α = 4 ε=0.01
stable fixed−pointunstable fixed−pointstable limit cycle
B
−1 0 1 2 3 4−1
−0.5
0
0.5
1
1.5
2α =4 ε=0.1
λ
v
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−1
−0.5
0
0.5
1
1.5
λ
v
α =4 ε=0.1
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
Figure 6: Bifurcation diagrams for the PWL1,2,1 model corresponding to Fig. 2 with ǫ = 0.01 (A) and
ǫ = 0.1 (B). For fixed-points we present their v-coordinate. For periodic orbits we present the maximum
and minimum values of their v-coordinate. The right panels are magnifications of the left ones.
16
A
−1 0 1 2 3 4−1
−0.5
0
0.5
1
1.5
2α =2 ε=0.1
λ
v
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−1
−0.5
0
0.5
1
1.5
λ
v
α =2 ε=0.1
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
B
−1 0 1 2 3 4 5−0.5
0
0.5
1
1.5
2
λ
v
α =2 ε=0.1
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
λ
v
α =2 ε=0.1
stable fixed−pointunstable fixed−pointstable limit cycleunstable limit cycle
Figure 7: Comparison between the bifurcation diagrams for the PWL1,2,1 model (A) and the corresponding
smooth FHN model (B) for α = 2 and ǫ = 0.1. For fixed-points we present their v-coordinate. For
periodic orbits we present the maximum and minimum values of their v-coordinate. The right panels are
magnification of the left ones.
17
where we use the index j to distinguish the dynamics of each linear regime. Here
Aj =
[
ηj −1
ǫα −ǫ,
]
bj =
[
−ηj vj−1 + wj−1
−ǫλ
]
. (17)
The solution to (16) may be written using matrix exponentials as
zj(t) = Gj(t)zj(0) +Kj(t)bj , Gj(t) = eAjt, Kj(t) =
∫ t
0Gj(s)ds. (18)
Since the phase space is divided naturally into only two pieces we may without loss of generality set
(v1, w1) = (0, 0). We construct a periodic orbit by evolving a trajectory according to (18), with initial
data (v,w) = (0, w(0)) (with w(0) as yet undetermined), until v = 0 is reached again for the first time
at T1 > 0. We then evolve the trajectory with new initial data (0, w(T1)) until meeting v = 0 again
a time T2 later. The “times-of-flight” Tj are determined by solving the threshold crossing conditions
v(T1) = 0 = v(T1 +T2). A periodic solution can then be found by solving w(T1+T2) = w(0) for w(0),
yielding the period T = T1 + T2. For the special case that the trajectory only visits R1 and R2, bjis independent of j and given by bj = −ǫ λ (0, 1)T . Evolution from initial data (0, w(0)) to (0, w(T1))
gives the pair of equations
G2(T1)
[
0
w(0)
]
− ǫλK2(T1)
[
0
1
]
=
[
0
w(T1)
]
. (19)
Dividing by ǫλ means that we may solve for this pair of equations (say using Cramer’s rule) in the
formw(0)
ǫλ= F1(T1),
w(T1)
ǫλ= F2(T1), (20)
for some explicit functions F1,2. A similar argument, with evolution of a trajectory from (0, w(T1)) to
(0, w(0)), gives
G1(T2)
[
0
w(T1)
]
− ǫλK1(T2)
[
0
1
]
=
[
0
w(0)
]
. (21)
Similarly we may solve for the pair (w(0), w(T1)) as
w(0)
ǫλ= F3(T2),
w(T1)
ǫλ= F4(T2), (22)
for some explicit functions F3,4. Equating (20) and (22) gives two simultaneous equations for the pair
(T1, T2):
F1(T1) = F3(T2), F2(T1) = F4(T2). (23)
which are independent of λ. Hence T = T1 + T2 is independent of λ.
3.3 The mechanism of generation of small and large amplitude limit
cycles and the abrupt transition between them
We begin by explaining the mechanisms that govern the generation of small and large amplitude limit
cycles and the abrupt transition between both limit cycle amplitude regimes in the context of the
18
example presented in Fig. 5 as λ changes from 0.029 to 0.03. The parameters we used (η1 = 0.3,
α = 4 and ǫ = 0.01 with η+cr = 0.39) correspond to the supercritical canard phenomenon illustrated in
Fig. 1-A for the smooth FHN system. As we progress in our discussion, we explain how changes in
the values of these parameters affect the resulting dynamics. We will focus on changes in the relative
lengths of L1 and L2, ǫ and α.
The dynamics of a PWL1,2,1 system is divided into four linear regimes Rj corresponding to the
four linear pieces Lj, and indexed accordingly (see Section 2.3) as shown in Fig. 2. The dynamics
in each of these regimes are governed by a linear system of the form (4). The initial conditions
for each regime Rj are equal to the values of v and w at the end of the previous regime (either
Rj−1 or Rj+1) as described in Section 2. Fig. 8-A shows the two trajectories in Fig. 5-A and -B
(λ = 0.029 and λ = 0.03) in the same phase-plane. (The v-nullcline is independent of λ. The w-
nullclines for both values of λ are indistinguishable. We plotted the w-nullcline corresponding to Fig.
5-A.) In these figures we have plotted the limit cycle trajectories after transients disappeared. These
trajectories have been computed using the analytical solutions presented in Section 2.3. Alternatively,
they can be computed numerically. Geometric and dynamic information corresponding to the four
linear regimes are summarized in Table 1. We will refer to this table in our explanation.
As we mentioned in Section 1, the analytical solutions for PWL systems provide a poor insight into
the models’ dynamics. Below we use dynamical systems tools to explain these dynamics and answer
various relevant questions about the abrupt transition between limit cycle amplitude regimes. In our
explanation we investigate how limit cycle trajectories initially in the linear regime Rl,1, near the left
branch of the v-nullcline Ll,1, return to this regime after a cycle. One difficulty in investigating these
limit cycle trajectories is that it is not possible to consider an initial point exactly on the limit cycle.
We overcome this difficulty by computing an asymptotic approximation (for small values of ǫ) to the
limit cycle trajectory in the linear regime Rl,1 (see Appendix B.4). This is the trajectory we follow in
our explanation and we will refer to it as the limit cycle trajectory although it is only an asymptotic
approximation to the “real” limit cycle trajectory.
Linear regimes: virtual and actual fixed-points
A distinctive feature of PWL systems is that in each regime Rj dynamics are organized around a
virtual fixed-point (vj , wj) which results from the intersection between the w-nullcline Nw(v) = α v−λ
and the linear piece Lj or its extension beyond the boundaries of Rj, [vj−1, vj ]. In the latter case,
virtual fixed-points are located outside the corresponding regime; i.e. vj /∈ [vj−1, vj ]. We illustrate
this in Fig. 8-B for the linear regime Rl,1. The vertical dashed lines separate between the linear
regimes Rl,1 and R1. The actual fixed-point (v, w) of the PWL1,2,1 system (blue dot) is located on
the intersection between the v- and w-nullclines (solid-red and -green lines respectively). The virtual
fixed-point (vl,1, wl,1) for the linear regime Rl,1 is located on the intersection between the extension of
the linear piece Ll,1 (dotted-red line) and the w-nullcline which occurs in the linear regime R1 (and
not in the linear regime Rl,1). Clearly, the actual fixed-points (v, w) of a PWL system is also the
virtual fixed-points for the regime where it is are located (see Fig. 9-B).
Within the boundaries of each regime, trajectories evolve as if the corresponding linear system,
19
which governs their dynamics as long as the trajectory is in that regime, govern their dynamics globally
(for all values of t). In other words, within the boundaries of each regime trajectories evolve according
to the linear dynamics defined in that regime and they “do not feel” that the dynamics governing
their evolution will change at a future time when the trajectory moves to a different regime.
The dynamics in the linear regime Rl,1
In Fig. 8-B we consider two trajectories initially located in the linear regime Rl,1, close to the
v-nullcline. The “blue” one is the limit cycle trajectory which evolves according to the PWL1,2,1
system. It evolves in a small neighborhood of the v-nullcline due to the fast-slow nature of the system
(ǫ = 0.01 ≪ 1). The “dotted-cyan” trajectory evolves according to the dynamics of the linear regime
Rl,1. It heads towards, and eventually converges to, the virtual fixed-point (vl,1, wl,1) (cyan dot) which
is a stable node. (This follows from the analytic solution presented in Section 2.3. Since ǫ ≪ 1, it
also follows from the asymptotic approximation to the slow manifold computed in Appendix B.4).
For as long as the “blue” trajectory is in the linear regime Rl,1 it evolves as the cyan one; i..e, as if
it were attracted to the virtual fixed-point (cyan dot). This virtual fixed-point ceases to be the blue
trajectory’s target once this trajectory crosses the boundary between the linear regimes Rl,1 and R1
since its dynamics is no longer governed by the linear regime Rl,1 but by the linear regime R1.
The dynamics in the linear regime R1
The dynamics of the trajectory in the linear regimeR1 is shown in Fig. 9. This figure is a magnification
of Fig. 8-A. The linear piece L1 has slope η1 = 0.3 < η+cr = 0.39. The dashed-red lines represent the
extension of the linear piece L1 beyond the boundaries of the linear regime R1. The corresponding
fixed-point (blue dot) is an unstable focus (see Table 1) located on the intersection between L1 (solid-
red line) and the w-nullcline (green line) and is located in the linear regime R1. We show three
trajectories in the right panel. The “solid-blue” and “dashed-blue” trajectories correspond to the
small and large amplitude limit cycles shown in Fig. 8-A for λ = 0.029 and λ = 0.03 respectively. The
“solid-blue” trajectory crosses the linear piece L1 “almost” at the boundary between the linear regimes
R1 and R2 and never enters R2. The “dashed-blue” trajectory crosses this boundary and moves into
the linear regime R2. We address the dynamics of this trajectory in next section. The “dashed-cyan”
trajectory initially coincides with the “solid-blue” trajectory and its evolution is governed (globally)
by the linear regime R1. The left panel shows this “cyan” trajectory in the absence of the “blue” ones.
Since the fixed-point is a focus, the “cyan” trajectory spirals out. (We only show the trajectory for
the relevant time interval.) Once it enters the linear regime R1, the “solid-blue” trajectory also spirals
out as if its dynamics were globally governed by this regime. The “solid-blue” and “cyan” trajectories
split when they cross from R1 to Rl,1. This has no consequences for the “cyan” trajectories which
continues to spiral out. The dynamics of the “blue” trajectory, instead, ceases to be governed by that
of R1 and returns to be governed by the linear regime Rl,1. Consequently, it crosses the v-nullcline
(solid-red line). The reminder of the dynamics are as explained above for the linear regime Rl,1 (see
Fig. 8).
20
A B
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
Nv(v)
Nw
(v)
trajectories
−0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−8
−6
−4
−2
0
2
4
6
8
10x 10
−3
v
w
Rl,1
R1
Figure 8: Canard-like phenomenon in a PWL1,2,1 model with α = 4 and ǫ = 0.01. Dynamics in
the linear regime Rl,1. The nullclines are as in Fig. 2. The vertical (thin) dashed-lines separate between
linear regimes and intersect the v-nullcline Nv at the joint points between the corresponding linear pieces.
The actual fixed-point of the PWL1,2,1 system is located in the linear regime R1 and indicated with a blue
dot (panel B) on the intersection between the two nullclines. A. Super-imposed trajectories for the small
(solid) and large (dashed) amplitude limit cycles showed in Figs. 5-A and -B for λ = 0.029 and λ = 0.03
respectively. B. Dynamics of the trajectory in the linear regime Rl,1. The dotted-red line indicates the
continuation of the linear piece Ll,1 beyond the boundaries of the linear regime. The virtual fixed-point
(vl,1, wl,1) = (0.0058,−0.0058) for Rl,1 is located in R1 and indicated with a cyan dot (intersection between
the dotted-red line and the w-nullcline Nw). The dotted-cyan curve corresponds to a trajectory whose
global dynamics are governed by the linear regime Rl,1 and shows that the virtual fixed-point is a stable
node. Initially, this trajectory is located near the trajectory of the PWL1,2,1 system (solid-blue). The slope
of the linear pieces in Nv (from left to right) are: ηl,1 = −1 (left branch), η1 = 0.3 and η2 = 1.3 (middle
branch), and ηr,1 = −1 (right branch). The linear pieces in the middle branch join at (v1, w1) = (0.3, 0.09).
21
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.05
0
0.05
0.1
0.15
0.2
v
w
Rl,1
R1
R2
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.05
0
0.05
0.1
0.15
0.2
v
w
Rl,1
R1
R2
Figure 9: Canard-like phenomenon in a PWL1,2,1 model with α = 4 and ǫ = 0.01. Dynamics in
the linear regime R1. This figure is a magnification of Fig. 8-A. The nullclines are as in Fig. 2. The
solid-red PWL curve and the solid-green line represent the v- and w-nullclines (Nv and Nw) respectively.
The vertical (thin) dashed-lines separate between linear regimes and intersect Nv at the joint points
between the corresponding linear pieces. The dotted-red lines represent the continuation of the linear
piece L1 beyond the boundaries of the linear regime. The actual fixed-point (v, w) = (0.0078, 0.0024) of
the PWL1,2,1 system is located in the linear regime R1 and indicated with a blue dot on the intersection
between the two nullclines. The dotted-red lines indicates the continuation of the linear piece L1 beyond
the boundaries of the linear regime. The “dotted-cyan” curve corresponds to a trajectory whose global
dynamics are governed by the linear regime R1 and illustrates that the fixed-point is an unstable focus.
The “cyan” trajectory spirals out with frequency µ = 0.126. The slope of the linear pieces in Nv (from left
to right) are: ηl,1 = −1 (left branch), η1 = 0.3 and η2 = 1.3 (middle branch), and ηr,1 = −1 (right branch).
The linear pieces in the middle branch join at (v1, w1) = (0.3, 0.09).
22
The small amplitude limit cycle in Figs. 8 and 9 corresponds to a value of λ (= 0.029) for which
the limit cycle trajectory intersects the v-nullcline in the linear regime R1 and never crosses to the
linear regime R2. Limit cycle trajectories for smaller values of λ intersect the linear piece L1 at lower
values of v (not shown). These values of v depend on the amplitude of the initial oscillation of the
trajectory after entering the linear regimes R1 (see solid-blue trajectory in Fig. 9). This amplitude,
in turn, depends on the distance
D = D(w0, λ) = D(w0, v1, w1) =√
(v0 − v1)2 + (w0 − w1)2 (24)
between the initial point (v0, w0) = (0, w0) in R1 and the fixed-point (v1, w1) ∈ R1, parametrized
by λ, through the constants c1 and c2 (11) in the solution (10) presented in Section 2.3. (This
parametrization is well defined since we assumed that for each value of λ there is a unique fixed-
point.) Note that in (24) v0 = 0 and w1 = η1 v1. To leading order, the initial value w0 can be
analytically approximated using the asymptotic computation in Appendix B.4. For ǫ ≪ 1, w0 = O(ǫ).
For λ = 0, D = 0 since (v1, w1) = (0, 0) is a stable node. In Section 3.5 we show that D is an
increasing function of λ. For a fixed value of the length of the linear piece L1, if D is small enough,
then the trajectory crosses L1 as it occurs in Figs. 8 and 9 for the “solid-blue” trajectory. On the
other hand, if D, and hence λ, is large enough, then the trajectory reaches the boundary between
the linear regimes R1 and R2 without crossing the linear piece L1 as it occurs in Figs. 8 and 9 for
the “dashed-blue” trajectory. Once the trajectory crosses this boundary its dynamics ceases to be
goverend by the linear regime R2 and turns to be governed by the linear regime R2.
The dynamics in the linear regime R2
The dynamics of the trajectory in the linear regime R2 is shown in Fig. 10. The linear piece L2 has
slope η2 = 1.3 > η+cr = 0.39. The corresponding virtual fixed-point (blue dot) is an unstable node
(see Table 1) located on the intersection between the extension of the linear piece L2 (dashed-red
lines) and the w-nullcline (green line) and is located in the linear regime Rl,1 (not in R2). As in the
previous figures, we show three types of trajectories. The “solid-blue” and “dashed-blue” trajectories
correspond to the small and large amplitude limit cycles shown in Fig. 8-A for λ = 0.029 and
λ = 0.03 respectively. Their dynamics are governed by the PWL1,2,1 system. Both evolve very close
in the linear regimes Rl,1 and R1 and bifurcate near the boundary between these two linear regimes.
The two “dashed-cyan” trajectories evolve according the dynamics of the linear regime R2 for all
values of t. They are initially located near the virtual fixed-point and the “dashed-blue” trajectory
respectively. They move fast along almost horizontal direction as expected from fast-slow systems.
They illustrate the fact that the virtual fixed-point is an unstable node (see Table 1, set II).
The “dashed-blue” trajectory eventually crosses the boundary between the linear regimes R2 and
Rr,1 and its dynamics changes accordingly.
The dynamics in the linear regime Rr,1
The dynamics of the trajectory in the linear regime Rr,1 is shown in Fig. 11. It is qualitatively
similar to the dynamics on the linear regime Rl,1. The linear piece Lr,1 has slope ηr,1 = −1. The
23
−1 −0.5 0 0.5 1 1.5 2
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
v
w
R1R
l,1 Rr,1
R2
Figure 10: Canard-like phenomenon in a PWL1,2,1 model with α = 4 and ǫ = 0.01. Dynamics
in the linear regime R2. The nullclines are as in Fig. 2. The solid-red PWL curve and the green
line represent the v- and w-nullclines (Nv and Nw) respectively. The vertical (thin) dashed-lines separate
between linear regimes and intersect Nv at the joint points between the corresponding linear pieces. The
dotted-red lines represent the continuation of the linear piece L2 beyond the boundaries of the linear
regime. The virtual fixed-point (v2, w2) is located in Rl,1 and indicated with a blue dot (intersection
between the dotted-red line and the w-nullcline Nw). The dashed-cyan curves correspond to trajectories
whose global dynamics are governed by the linear regime R2 and illustrate that the fixed-point is an unstable
node. Initially, the two cyan curves are located near the virtual fixed-point and near the trajectory of the
PWL1,2,1 system (solid-blue curve) respectively. The slope of the linear pieces in Nv (from left to right)
are: ηl,1 = −1 (left branch), η1 = 0.3 and η2 = 1.3 (middle branch), and ηr,1 = −1 (right branch). The
linear pieces in the middle branch join at (v1, w1) = (0.3, 0.09).
24
corresponding virtual fixed-point (blue dot) is located on the intersection between the extension of the
linear piece Lr,1 (dashed-red line) and the w-nullcline (green line) and is located in the linear regime
R2. In the left panels we show the trajectories of the PWL1,2,1 system (using the same notation).
For clarity, in the right panel we show only trajectories whose dynamics are governed (globally) by
the linear regime Rr,1. This illustrates that the virtual fixed-point is a stable node (see also Table 1,
set I). The “dashed-blue” trajectory crosses the linear piece Lr,1 and continues to evolve in a close
vicinity of it heading towards the virtual fixed-point. When the trajectory crosses the boundary
between the linear regimes Rr,1 and R2 its dynamics ceases to be governed by the linear regime Rr,1
and, consequently, the virtual fixed-point ceases to be the trajectory’s target. The reminder of the
dynamics of the limit cycle trajectory is standard for relaxation oscillators and can be explained using
the same ideas explained above.
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
Rl,1
R1
R2 R
r,1
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
wR
2 Rr,1
R1R
l,1
Figure 11: Canard-like phenomenon in a PWL1,2,1 model with α = 4 and ǫ = 0.01. Dynamics in
the linear regime Rr,1. The nullclines are as in Fig. 2. The vertical (thin) dashed-lines separate between
linear regimes and intersect the v-nullcline Nv at the joint points between the corresponding linear pieces.
The dotted-red lines indicates the continuation of the linear piece Lr,1 beyond the boundaries of the linear
regime. The virtual fixed-point (vr,1, wr,1) is located in R2 and indicated with a blue dot (intersection
between the dotted-red line and the w-nullcline Nw). The dotted-cyan curves correspond to trajectories
whose global dynamics are governed by the linear regime Rr,1 and illustrate that the fixed-point is a stable
node. Initially, the cyan curves are located near and away from the trajectory of the PWL1,2,1 system
(solid-blue) respectively. The slope of the linear pieces in Nv (from left to right) are: ηl,1 = −1 (left
branch), η1 = 0.3 and η2 = 1.3 (middle branch), and ηr,1 = −1 (right branch). The linear pieces in the
middle branch join at (v1, w1) = (0.3, 0.09).
Together, the results in this section provide a geometric and dynamic picture of how the canard-
like explosion occurs in PWL1,2,1 systems that can be extended to other parameter regimes. To a first
approximation, a small amplitude limit cycle is created if the amplitude of the initial spiraling out
25
oscillation in the linear regime R1 is small enough for the trajectory to be able to cross the linear piece
L1 and a large amplitude oscillation is created otherwise. This initial amplitude depends on both λ
and the properties (length and slope) of the linear piece L1. The dependence of λ occurs through
the distance between the trajectory’s initial point in the linear regime R1 and the fixed-point. For
all other parameters fixed, an increase in the value of λ causes increase in this initial distance and, in
turn, causes the amplitude of the small amplitude limit cycle to increase until, for some critical value
of λ, the trajectory cannot longer cross the linear piece L1. We analyze the effects of changes in the
model parameters on the canard-like phenomenon later in this paper.
An important aspect of these dynamics, highlighted by our results, is that trajectories evolve in
a close vicinity of the unstable (middle) branch of the cubic-like function for a significant amount
of time, as it occurs for smooth systems, not because they are attracted to it nor because they are
not repelled enough. They are indeed repelled but in a spiraling-out manner. For small amplitude
oscillations, this repulsion ceases when the trajectory returns to the vicinity of the slow manifold (left
branch). For large amplitude oscillations, the repulsion continues once the trajectories enter the linear
regime R2 but it changes from “focus-like” to “node-like” which enables the trajectory to move fast
away from the fixed-point towards the right branch of the PWL function.
Inflection lines and eigenvectors
For ǫ ≪ 1, between small and large amplitude limit cycles there exists an orbit with a point of inflection
[35, 36, 37] (see Appendix B.5). This generates a curve (line) in phase-plane dividing between regions
of small and large amplitude limit cycle oscillations as we show in Fig. 5-C. We derive the inflection
lines Linfl in Appendix B.5. For the linear system governing the dynamics in R2 the inflection lines
are given by
L1,2infl(v) = α
η − r1,2 − ǫ
α− r1,2 − ǫv +
λ (r1,2 + ǫ)− α (η v − w)
α− r1,2 − ǫ(25)
where r1,2 are the eigenvalues (6) of the system in R2, η is the slope of L2 and (v, w) are the left
end-point of L2 (v1, w1). In Fig. 5-C we present the inflection lines corresponding to the transition
between the small and large amplitude limit cycles (λ = λc). In Appendix B.6 we show that the
inflection lines coincide with the “eigenvector lines” W2,1(v)
W2,1(v) = ( r2,1 + ǫ ) v − ( r2,1 + ǫ ) v + w (26)
that have the eigenvector directions and cross the virtual fixed-point (v2, w2) of the system. Note that
for ǫ = 0 both lines coincide with the v-nullcline.
3.4 The canard critical value λc
In smooth systems, the canard phenomenon occurs for a range of values of λ which is exponentially
small in ǫ. Within this small critical range, as λ increases, the amplitude of the limit cycle increases
between these values corresponding to the small and large amplitude oscillations regimes, and the
26
limit cycle trajectory is able to move in a close vicinity of the unstable branch of the v-nullcline for
a significant amount of time (see Fig. 1-A). There is a value of λ within this critical range that
corresponds to the so called maximal canard. This value of λ is typically used to define the canard
critical value λc. However, maximal canards are typically not observable in realistic situations and the
exact value of λc is difficult to compute. Approximations to λc have been obtained (see discussion in
Appendix A.2). These approximations can be thought of as approximations to the critical range over
which the abrupt transition occurs, and are useful to predict whether a smooth system will display
either small or large amplitude oscillations with a high degree of accuracy.
Following these ideas for smooth systems, we provide here a working definition of the canard
critical value λc for PWL systems as the value of λ separating between small and large limit cycle
amplitude regimes, and we provide a method for approximating λc, and the critical range over which
the abrupt transition occurs, that allows to make quantitative predictions.
For the parameters in Fig. 5 (and Figs. 8 to 11), to a first approximation, limit cycles have small
amplitude for λ ≤ 0.029 and large amplitude for λ ≥ 0.03. In addition, λ = 0.029 has been graphically
identified as the“last” value of λ for which the limit cycle trajectory crosses the linear piece L1; i.e.,
the intersection between the limit cycle trajectory and the v-nullcline approximately occurs at the
joint point between the linear pieces L1 and L2, and a very small increase in the value of λ generates
a large amplitude limit cycle. A closer examination shows that small amplitude limit cycles can cross
the linear piece L2 for values of λ ∈ (0.029, 0.02931) as we show in Fig. 5-C. However, the range of
values of λ for which this occurs (0.00031) is small as compared to λ = 0.029, so choosing λ = 0.029 as
the critical value of λ for which the abrupt transition between small and large amplitude limit cycles
occurs (canard-like phenomenon) provides a good approximation to the “real” critical value defined
above.
Generalizing these ideas, we approximate the canard critical value by the value of λ for which the
limit cycle trajectory intersects the middle branch of the v-nullcline at the joint point between the
linear pieces L1 and L2. (This is the “last” value of λ for which limit cycle trajectories do not cross
from R1 to R2.) Similarly to the approximations to the canard critical value for smooth systems
discussed above (and in Appendix A), we will use the symbol λc and the term canard critical value to
refer to this approximation. For the parameters in Figs. 5, comparison of this value (λc = 0.029) with
the more accurate one in Fig. 5-C (λ = 0.0293125) where small amplitude limit cycle trajectories are
able to cross to R2 yields a relative error equal to 0.01069 (1.069%). (The value λ = 0.0293125 has
been numerically calculated as the mean between the values of λ corresponding to the “last” small
and “first” large amplitude oscillations. Note that the relative error is of the same order of magnitude
as ǫ = 0.01.)
The canard critical value λc increases with the size of L1
We illustrate this in Fig. 12 where all the parameters are the same as in Fig. 5 with the exception
of |L1|, that is larger in Fig. 12 than in Fig. 5. In particular, the slope η1 = 0.3 is the same in Figs.
12 and 5. From eq. (15), the slope of L2 (η2 = 1.467) is larger in Fig. 12 than in Fig. 5. From our
discussion in Section 3.1, the stability properties of the virtual fixed-points in both R1 and R2 do not
27
change as |L1| increases; they are an unstable focus and unstable node regimes respectively. (Note
that since α and ǫ are the same for both figures so is η+cr).
In Fig. 12-A, λ = 0.03. In contrast to Fig. 5-B, the trajectory is able to cross the linear piece
L1 in Fig. 12-A because of the increased length of L1. Thus the system has a small amplitude limit
cycle for a value of λ for which Fig. 5-A shows a large amplitude limit cycle. Only when λ increases
to λ = 0.039, corresponding to (v1, w1) = (0.0105, 0.0032), does the canard explosion occur. Similarly
to Fig. 5, the virtual fixed-point (v2, w2) = (−0.17,−0.71) in R2 is an unstable node so the trajectory
moves away from R1 and generates a large amplitude limit cycle (Fig. 12-B). The canard explosion
in this case occurs for values of λ ∈ (0.038, 0.039) and can be approximated by λc = 0.038. As λ
increases further, D(w0, λ) also increases, and the limit cycle approaches a fully developed relaxation
oscillator (Fig. 12-C).
These ideas can be generalized. For a given set of parameters (α, ǫ, η1), λc depends on the initial
amplitude D(w0, λ) of the limit cycle trajectory in R1, given by eq. (24), and |L1|. In Appendix B.4
we show that for ǫ ≪ 1, to a first approximation, initial conditions in R1 are independent of λ (they
differ in O(ǫ)). Then, for ǫ ≪ 1, the initial distance D(w0, λ) is independent of w0, and λc depends
only on the location of the fixed point (v1, w1), parametrized by λ (14), and |L1|. As λ increases,
the fixed-point moves to the right, D(w0, λ) increases, and the limit cycle trajectory intersects the
v-nullcline Nv at a “higher” point until this intersection no longer occurs in R1. As before, we take an
approximation to λc as the maximum value of λ for which the trajectory intersects the linear piece L1
at its joint point with the linear piece L2. For all other parameters fixed, as |L1| increases, the range
of values of λ for which limit cycle trajectories and Nv intersect in R1 also increases, and so does λc.
In principle, the mechanism described here works for linear pieces L1 with arbitrary lenghts |L1| <(1 + η21)
1/2 (within the boundaries of the middle branch). However, the technically small amplitude
limit cycles created by long linear pieces L1 may not lead to small amplitude oscillations; i.e., to
oscillations whose amplitude is an order of magnitude smaller than these created by large amplitude
limit cycles. We illustrate this in Fig. 12-D and -E for a linear piece L1 with right endpoint v1 = 0.9.
The canard-like explosion still occurs for λ ∈ (0.0868, 0.0869) but the limit cycle amplitudes in both
regimes are roughly of the same order of magnitude.
Effects of changes in ǫ on the canard-like phenomenon and λc
The canard phenomenon described above occurs because limit cycle trajectories are able to cross the
middle (unstable) branch of the v-nullcline for values of λ < λc while they are not able to do so for
larger values of λ. This is the result of two different features of the system: the time scale separation
(ǫ ≪ 1) and the cubic-like nonlinearities, in particular the fact that the middle branch breaks into two
linear pieces with different stability properties, unstable focus (R1) and node (R2) respectively. The
time scale separation is responsible for “forcing” trajectories to move away from the linear piece L2
(in the linear regime R2) thus preventing trajectories from crossing L2 to create a small amplitude
limit cycle. The result is a large amplitude limit cycle.
In order to further understand the effect of time scale separation on the canard-like phenomenon,
we show in Fig. 13 the phase-diagrams for two representative values of ǫ (larger than the one we
28
A B
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.03
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.039
Nv(v)
Nw
(v)
C
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.2
Nv(v)
Nw
(v)
D E
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.0868
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.0869
Nv(v)
Nw
(v)
Figure 12: Canard phenomenon in a PWL1,2,1 model of FHN type. We used η1 = 0.3, ǫ = 0.01
and the parameters presented in Table 1 (sets I, III and IV). A Small amplitude limit cycle for λ = 0.03,
v1 = 0.4 and the parameters presented in Table 1 (sets I and III). B Large amplitude limit cycle for
λ = 0.039, v1 = 0.4 and the parameters presented in Table 1 (sets I and III). C Large amplitude limit cycle
for λ = 0.2, v1 = 0.4 and the parameters presented in Table 1 (sets I and III). D Small amplitude limit
cycle for v1 = 0.9, λ = 0.0868 and the parameters presented in Table 1 (sets I and IV). E Large amplitude
limit cycle for v1 = 0.9, λ = 0.0869 and the parameters presented in Table 1 (sets I and IV).
29
considered so far): ǫ = 0.1 (panels A) and ǫ = 0.3 (panels B). For ǫ = 0.1, although the time scale
separation is smaller than for ǫ = 0.01, the transition between amplitude regimes is still abrupt. It
occurs for values of λ ∈ (0.385, 0.3852). Although this interval is larger than (0.02931, 0.029315) (the
corresponding interval for ǫ = 0.01), it is still very small. In this case, λc = 0.385 which is larger than
the critical value for ǫ = 0.01. As expected, the shapes of the limit cycles differ from those in Fig. 5.
The limit cycle trajectories evolve further away from the v-nullcline than for ǫ = 0.01 and the large
amplitude limit cycle trajectory does not move almost horizontally in the linear regime R2 as happens
for ǫ = 0.01. The stability properties for the two cases (ǫ = 0.01 and ǫ = 0.1) have similarities and
differences. Similarly to the ǫ = 0.01 case, the fixed-points for ǫ = 0.1 in the linear regimes R1 and R2
(middle branches) are unstable foci and nodes respectively (not shown). In contrast to the ǫ = 0.01
case, the virtual fixed-points in the linear regimes Rl,1 and Rr,1 (left and right branches respectively)
are stable focus rather than stable nodes (not shown). This does not qualitatively change the basic
mechanism explained in Section 3.3.
For ǫ = 0.3 (Fig. 13-B), the transition from small to large amplitude limit cycles is much less
abrupt and occurs on a larger range of values of λ over which trajectories are able to cross the linear
piece L2. Associated with the abrupt transition in Fig. 13-A for ǫ = 0.1, there is the inflection of the
trajectory as it crosses from the linear regime R1 to R2. This inflection is absent for ǫ = 0.3 in Fig.
13-B.
3.5 Analytical approximation of the canard critical point
Here we show that given α, ǫ ≪ 1, η1 (slope of L1), v1 (joint point between L1 and L2), and the
initial value w0, one can predict to a good degree of accuracy whether the PWL1,2,1 system will have
either a small or large amplitude limit cycle for each value of λ, and consequently one can compute
an approximate value for λc. In the linear regime R1, the left end-point of L1 is (v0, w0) = (0, 0) and
hence the initial condition is given by (v0, w0) = (v0, w0) = (0, w0). In our simulations we have used
the asymptotic approximation w0 = −λ ǫ (for small values of ǫ) computed in Appendix B.4 (see also
[2]). This allows to approximate the initial point in the limit cycle trajectory without knowledge of
the trajectory in the previous regime. We show below that the choice of other (small) values for w0
do not qualitatively alter the picture described here.
As we mentioned earlier in the paper, changes in λ do not affect the stability properties of the
corresponding virtual fixed-point (v1, w1) but only its location on the linear piece L1 (or the extension
of L1 beyond the boundaries of R1) through the reparametrization given by (14). These changes in
the location of the fixed-point (v1, w1) cause changes in the integration constants c1 and c2 given by
(9) (focus case) and (11) (node case). Note that c1 and c2 depend on λ only through (v1, w1) or,
alternatively, through the initial conditions referred to the virtual fixed point (v0 − v1, w0 − w1).
As a result, neither the frequency µ nor the amplitude coefficient r (exponential factor in the
solution (10)) change with λ, but rather as λ incresases the spiraling-out trajectory is initially in a
larger amplitude “orbit”, measured by D = D(w0, λ) given by (24). We illustrate this in Fig. 14-A
for the linear piece L1 (red line) with slope η1 = 0.3 and the other parameters used in Figs. 5 and 12
(α = 4 and ǫ = 0.01). Trajectories corresponding to increasing values of λ intersect L1 and reverse
30
A
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4 λ = 0.385
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
vw
α = 4 λ = 0.3852
Nv(v)
Nw
(v)
B
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4 λ = 0.55
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4 λ = 0.58
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4 λ = 0.59
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
v
w
α = 4 λ = 0.65
Nv(v)
Nw
(v)
Figure 13: Transition from small to large amplitude limit cycles in a PWL1,2,1 model of FHN
type. We used v1 = 0.3, η1 = 0.3, and the parameters presented in Table 1 (sets I and II). A ǫ = 0.1. B
ǫ = 0.3.
31
their direction at the reversing, or turning, points (vrev, wrev) whose coordinates increase with λ (the
reversing points move to the right). In Fig. 14-B we plot vrev as a function of λ for various values
of η and α. The green line in the left panel corresponds to the parameters in Fig. 14-A. This figure
shows that the dependence of vrev with λ is linear for all values of η (< η+cr) and α considered; i.e.,
vrev = v(trev) = κ(η, ǫ, α) λ, (27)
and that the slope κ increases with increasing values of η and decreasing values of α.
The curves vrev vs. λ have been calculated using the following formulas derived in Appendix B.2:
vrev = v(trev), wrev = w(trev) (28)
with
trev =1
µtan−1 c2 µ+ c1 r
c1 µ− c2 r, (29)
where c1, c2, µ, r are given by (11), (12) and (13), and depend on α, ǫ, η1 and λ. If for a given
parameter set α, ǫ and λ, and η1 and v1 (which determine |L1|) the point (vrev, wrev) ∈ L1 (vrev ≤ v1),
then the trajectory crosses L1 and a small amplitude limit cycle is created. If, on the other hand,
vrev > v1, then the trajectory moves into regime R2 and, to a first approximation, a large amplitude
limit cycle is created.
The values of λ for which a limit cycle trajectory crosses L1 are bounded by λ < v1/κ since
vrev < v1. The larger κ, the smaller the range of values of λ for which trajectories can cross L1 and
create small amplitude limit cycles. Following our previous ideas, we take
λc = v1/κ (30)
as an approximation to the canard critical value. For the parameters in Fig. 5-B λc = 0.029. This
approximation assumes that R2 does not support small amplitude limit cycles limit cycles; i.e., large
amplitude limit cycles are created vrev > v1. As we discussed above, although small amplitude limit
cycles trajectories are able to move away from R1 into R2 and cross L2 (Fig. 5-C), this occours for
a range of values of λ which is small as compared to λc. In the next sections we present appropriate
quantitative comparisons.
The linear dependence of vrev with λ given by eq. (27) can be analytically shown to occur in R1
if in addition to (v, w) = (0, 0) and v0 = 0 one uses the asymptotic approximation w0 = −λ ǫ (see
Appendix B.4). We show in Appendix B.2 that under these conditions eq. (29) becomes
trev =1
µtan−1 −2µ
2 + η − ǫ(31)
with
κ(η, ǫ, α) =2
α− η
−1− η + ǫ− ǫ α+ ǫ η
2 + η − ǫcos (µ trev) e
(η−ǫ) trev/2. (32)
Note that the slope κ is independent of λ.
32
Linear dependence of the turning points vrev with λ is maintained even if we lift the assumption
w0 = −λ ǫ although in this case we cannot obtain an analytic expression. We illustrate this in Fig.
15-B for w0 = −0.01 and w0 = −0.001. As w0 increases in absolute value (the initial point in R1
is further below the minimum of the v-nullcline), vrev increases thus favoring the transition to R2.
Similar arguments to these presented here can be applied to models having a larger number of linear
pieces as we will see in Section 4.
Dependence of λc with other model parameters
As Fig 14-B illustrates, the slopes κ increase with η (left panel) and decrease with α (comparison
between left and right panels), and so λc (= v1 /κ) decrease with η and increase with α. In other
words, for fixed values of ǫ, increases in the value of η and decreases in the value of α favor the
“escape” from R1 without crossing the v nullcline, and so the creation of large amplitude limit cycles
for smaller value of λ.
The dependence of κ with η is not linear as eq. (32) and Fig. 14-B show. In Fig. 15-A we present
the curves κ(η) for various values of ǫ and two values of α: α = 4 (left panel) and α = 2 (right panel).
The domains of these graphs are bounded by η+cr (7) in order to keep R1 as an unstable focus regime.
The curves κ(η) are asymptotic to the vertical lines η = η+cr (not shown). As we can see in Fig. 15-A,
κ is a decreasing function of ǫ. From our discussion in the previous paragraphs, λc decreases with
increasing values of κ, and hence is a decreasing function of η and an increasing function of both α and
ǫ, consistent with the canard critical value for smooth FHN systems given by eq. (54) in Appendix A.
In Fig. 16-A we show the predicted values of λc as a function of η for two values of α (α = 4 and
α = 2) and two values of ǫ (ǫ = 0.01 and ǫ = 0.05).
For certain parameter regimes it is possible for bistable periodic behavior to occur. One example
of this is shown in Fig. 6A. In this case an unstable periodic orbit exists that separates large and
small orbits. Under parameter variation this unstable medium size orbit is killed in a saddle-node of
periodics bifurcation. When it collides with the large orbit this must occur when two orbits, one with
v ≥ 1 and the other with v ≤ 1, collide. It is thus interesting to track the point in parameter space
where an unstable orbit passes through v = 1 and is maximal (which means that the orbit passes
through the right hand knee of the voltage nullcline). An example is shown in Fig. 17. Likewise we
could consider an unstable medium size orbit that collides with a small stable orbit, which amounts to
tracking the small stable orbit which passes through (v,w) = (v, w) as shown in the inset of Fig. 17.
Similar constructions have also been explored in [38]. However, to delineate canard transitions it is
useful to consider tracking orbits which pass through points of inflection on sections where v = v.
Curves of this type are also plotted in Fig. 17 and we see that the combination of these curves defines
a tongue of small but finite width, emanating from (λ, ǫ) ∼ (0, ǫc), that separates small from large
amplitude oscillations. Here ǫc is the critical value above which the inflection line exists, and can be
determined as a condition on the eigenvalues r1,2 in equation (5) to be real (see also Appendix B.5).
This gives ǫc = 2α − η2 − 2√
α(α − η2). Interestingly large amplitude unstable periodic solutions on
the branch that tracks those which pass through a point of inflection can have a shape exhibiting that
of a so-called maximal canard. An example of such an orbit is shown in Fig. 17 (inset bottom right)
33
A
0 0.2 0.4 0.6 0.8 1−0.1
0
0.1
0.2
0.3
0.4
0.5
v
w
η = 0.3 α = 4
Nv(v) = η v
λ = 0.08
λ = 0.07
λ = 0.09
λ = 0.06
λ = 0.05
λ = 0.04
λ = 0.03
λ = 0.02
λ = 0.01
B
0 0.02 0.04 0.06 0.08 0.10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 4
η = 0.1η = 0.2η = 0.3
0 0.02 0.04 0.06 0.08 0.10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 2
η = 0.1η = 0.2
Figure 14: A Nine Spiraling out trajectories different amplitudes. They correspond, from smaller to
larger amplitude, to λ = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09. Each trajectory starts at (v, w) ∼(0,−λ ǫ) and intersects the linear piece of the v-nullcline Nv(v) at points called (vrev, wrev). Parameters:
α = 4, η = 0.3, r = (η − ǫ)/2 = 0.145, µ = 0.126. B Curves of vrev as a function of λ for various values of
η. Left panel: α = 4 and ǫ = 0.01. Right panel: α = 2 and ǫ = 0.01. Note that the v-coordinate of the
fixed point is v = λ/(α− η).
34
A
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
20
η
κ
α = 4
ε=0.01ε=0.02ε=0.05ε=0.08ε=0.1
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
η
κ
α = 2
ε=0.01ε=0.02ε=0.05ε=0.08ε=0.1
B
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 4
η = 0.1w
o = −0.001
η = 0.1w
o = −0.01
η = 0.2w
o = −0.001
η = 0.2w
o = −0.01
η = 0.3w
o = −0.001
η = 0.3w
o = −0.01
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 2
η = 0.1w
o = −0.001
η = 0.1w
o = −0.01
η = 0.2w
o = −0.001
η = 0.2w
o = −0.01
C
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
v1
η 2 η1 = 1
η1 = 0.8
η1 = 0.5
ηcr
(α = 4)
ηcr
(α = 2)
η1 = 1.5
η1 = 0.1
η1 = 1.2
Figure 15: A Slope κ(η, ǫ, α) of the turning point lines in (27) as a function of η (slope of the linear piece)
for various values of α and ǫ. B Reversing (or turning) points vrev as a function of λ for various values
of η and α, and various initial points (0, w0) in R1. C Slope η2 of the linear piece L2 as a function of
v-coordinate v1 of the joint point between the linear pieces L1 and L2 for various values of the slope η1 of
the linear piece L1 and α.
35
(A)
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
0.4
0.5
η
λ c
α=4 ε=0.01α=4 ε=0.05α=2 ε=0.01α=2 ε=0.05
(B)
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
2
4
6
8
10
12x 10
−3
η
Eab
s,1
α=4 ε=0.01α=4 ε=0.05α=2 ε=0.01α=2 ε=0.05
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
η
Ere
l,1
α=4 ε=0.01α=4 ε=0.05α=2 ε=0.01α=2 ε=0.05
(C)
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
2
4
6
8
10
12x 10
−3
η
Eab
s,2
α=4 ε=0.01α=4 ε=0.05α=2 ε=0.01α=2 ε=0.05
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
η
Ere
l,2
α=4 ε=0.01α=4 ε=0.05α=2 ε=0.01α=2 ε=0.05
Figure 16: (A) Analytical values of λc as a function of η for various values of α and ǫ. (B) Absolute and
relative error between λc and the corresponding numerical approximations to the maximum values of λ for
which the limit cycle trajectory does not cross from R1 to R2. (C) Absolute and relative error between
λc and the corresponding numerical approximations to the maximum values of λ for which the limit cycle
trajectory crosses from R1 to R2 but still is a small amplitude limit cycle.
36
0
0.04
0.08
0.12
0.16
0 0.1 0.2 0.3 0.4
ε
λ
small
oscillations
large oscillations
bistability
Figure 17: Two parameter bifurcation of the PWL system for Fig. 7. For (λ, ǫ) ∼ (0, ǫc) we see a tongue
structure that separates large from small amplitude oscillations. The canard point resides within this
narrow tongue. For small λ the upper curve is defined by a small amplitude orbit passing through the
point (v, w) shown as a filled black circle in the inset. This line separates small amplitude from medium
amplitude orbits. The lower curve tracks orbits that pass through a point of inflection on the section v = v.
With increasing λ these orbits increase in size and the branch of solutions ultimately connects with the
branch of large amplitude periodic orbits that pass through the right hand knee of the voltage nullcline.
These unstable orbits can have shapes like those of maximal canards. For larger values of λ (∼ 0.32)
bistability of periodic solutions can occur. The horizontal line shows the value of ǫc, above which lines of
inflection can exist (a necessary condition for canard solutions to exist).
37
where we also plot the line of inflection (in solid green). The maximal canard trajectory follows the
inflection line for part of its trajectory and passes close to the right hand knee of the voltage nullcline.
Smaller amplitude orbits that pass through an inflection point occur for smaller values of λ, and this
branch of solutions is very close to that defining the border between small stable and medium unstable
orbits.
Accuracy of the analytical approximation
In order to check the accuracy of the analytical predictions λc = v1 /κ described above we compare
them with the numerical approximations to both the maximum value of λ for which the limit cycle
trajectory does not cross from R1 to R2 and the maximum value of λ for which the limit cycle
trajectory crosses fromR1 toR2 but still is a small amplitude limit cycle. We call these approximations
λc,1 and λc,2 respectively. We define the absolute and relative errors as follows
Eabs,1 = |λc − λc,1 |, Eabs,2 = |λc − λc,2 |, (33)
and
Erel,1 =Eabs,1
λc, Erel,2 =
Eabs,2
λc. (34)
Our results are presented in Figs. 16-B and -C. The analytical prediction λc = v1 /κ described above
provides a good approximation to the more accurate, numerically calculated, canard critical value
λc,2.
3.6 No small amplitude limit cycles exist when R1 is an unstable
node regime
No small amplitude limit cycles exist in R1 when it is an unstable node regime
In contrast to the situation described above, if the linear piece L1 has slope η1 > ηcr, then the
fixed point (v1, w1) (both virtual and actual) is an unstable node. In this case, due to the time
scale separation, trajectories arriving to R1 move almost horizontally, along a fast direction, and no
small amplitude limit cycles are generated; i.e., the PWL system displays relaxation (large amplitude)
oscillations for all values of λ. As in the focus case, in order for the trajectories to display small
amplitude oscillations, they should be able to intersect the linear piece L1 of the v-nullcline. We show
in Appendix B.3 that for nodes these intersection points can be calculated using eqs. (28) and
trev =1
r1 − r2ln
−c2 r2c1 r1
, (35)
where r1, r2, c1 and c2 are given by (6) and (9). Since r1 − r2 =√
(η + ǫ)2 − 4α ǫ > 0, in order for
trev to be positive (a necessary condiction for the trajectory to intersect L1) the following condition
has to be satisfied:
38
Q = −c2 r2c1 r1
=(v0 − v) (r2 + ǫ)− (w0 − w)
(v0 − v) (r1 + ǫ)− (w0 − w)
r2r1
> 1. (36)
We now show that Q < 1 in R1 for all values of λ; i.e., the trajectory never intersects the linear piece
L1 if the virtual fixed-point in R1 is a node.
The left-endpoint for L1 has coordinates v = w = 0 and the initial condition in R1 has v0 = 0.
From eq. 5, the fixed-point (both virtual and actual) is given by v = λ/(α − η) and w = λη/(α − η).
Substituting these values in (36), using eq. (6) and the facts that r1,2 + ǫ− η = −r2,1 and r1r2 = ǫ α,
and rearranging terms we get
Q(λ) =λ− q2λ− q1
(37)
with
q1 =w0 (α− η) r1
α ǫand q2 =
w0 (α− η) r2α ǫ
. (38)
Clearly, Q(0) = r2/r1 < 1. If there exist a value of λ > 0 such that Q(λ) = 1, then r2 = r1 which can
only occur if η = ηcr but η > ηcr. Alternatively, if there exist a λ > 0 such that Q(λ) > 1, then r1 < r2since w0 < 0, but r1 > r2 > 0. Note that our argument relies on the fact that w0 < w = 0. This is
always true in R1 but not in other regimes as we will see in Section 4 when we investigate a PWL1,3,1
system. Note also that, since dQ/dλ = w0 (α−η) (r2−r1) / (α ǫ) and r2−r1 = −√
(η + ǫ)2 − 4αǫ < 0,
then Q is an increasing function of λ for w0 < 0.
No small amplitude limit cycles exist in R2 when R1 is an unstable node regime
If R1 has an unstable node, then R2 is an unstable focus regime. The slopes η1 and η2 of R1 and
R2 respectively are related by eq. (15) where 0 < v1 < 1. (If v1 = 0, v1 = 1 or η1 = η2 = 1, then
the PWL system has only one linear piece in the middle branch; i.e., it is a PWL1,1,1 system, not a
PWL1,2,1 one.) The assumption in Section 3.1 that 0 < η1 < α imposes the following constraint on
η2:
η2 >1− α v11− v1
. (39)
Standard algebra shows that η2 = 1 for v1 = 0, and η2 is an increasing (decreasing) function of v1for η1 < 1 (η1 > 1). We show representative curves of η2 as a function of v1 according to eq. (15) in
Fig. 15-C. We also show the horizontal lines ηcr for α = 4 and α = 2, and ǫ = 0.01.
A necessary condition for R2 to have a focus is η2 < ηcr; i.e., η2 has to be below the line ηcr (red
line) for some value of v1 which together with η1 determines the length of the linear piece L1. For the
parameters in Figs. 6 (α = 4 and ǫ = 0.01), ηcr = 0.39. As discussed in Section 3.1 if R1 has a focus
(η1 < ηcr = 0.39), then R2 has a node since η2 is an increasing function of v1 for η1 < 1. For R2 to
have a focus, L1 has to be steeper than the line that joins the minimum and maximum of the cubic
PWL function (η1 > 1) but not as steep so as the endpoint of L1 is not higher than the maximum of
the PWL function. Alternatively, ηcr has to increase.
39
When L1 is a node and L2 is a focus only large amplitude limit cycles are created. As we explained
earlier, the trajectory arriving at R1 cannot cross the linear piece L1 and will move along the fast
(horizontal direction) arriving atR2 with a value of w not very far from its initial value in R1 (w0 < 0).
Since η2 < ηcr, from eq. (15)
w1 = η1 v1 > 1− ηcr (1− v1). (40)
That means that initially, the trajectory is not close enough to the point (v1, w1) (left endpoint of the
linear piece L2 so to be able to cross L2).
4 The canard phenomenon in a PWL1,3,1 models of FHN-
type
Here we extend the ideas discussed in Section 3 to PWL systems having three linear pieces in the
middle branch of the cubic-like function. More specifically, we consider PWL systems of FHN type
described by (2) where fpwl(v) is PWL1,3,1; i.e., it has three linear pieces (L1, L2 and L3) in its middle
branch as illustrated in Fig. 18-A. The left and right branches of the graph of fpwl(v) have a single
linear piece each, Ll,1 and Lr,1 respectively. The values of the slopes (η1, η2, η3, ηl,1 and ηr,1) are
presented in Table 2. The dynamics of this system is divided into five linear regimes corresponding
to the five linear pieces of fpwl(v) and indexed accordingly. The dynamics of the linear system within
each regime is governed by a linear component system of the form (4). The eigenvalues for each linear
regime are presented in Table 2. The dynamics on the regimes Rl,1 Rr,1 (left and right branches) are
as in the PWL1,2,1 models discussed in Section 3 (comparison between set I in Tables 1 and Table 2).
In Figs. 18-B, the limit cycle trajectory crosses the linear piece L1 at its right end-point. Figs.
18-C and -D illustrate the canard phenomenon as λ changes between 0.432 and 0.433. (Table 2, set II).
The endpoints (v1, w1) and (v2, w2) joining L1 to L2 and L2 to L3 respectively were choosen so that
the intervals [0, v1], [v1, v2] and [v2, 1] have equal size. The virtual fixed-point (v1, w1) is an unstable
focus while the virtual fixed-points (v2, w2) and (v3, w3) are unstable nodes.
The dynamics inR1 is qualitatively similar to the dynamics in the analogous regime in the PWL1,2,1
models discussed in Section 3 since the virtual fixed-point (v1, w1) is an unstable focus. For values of
λ such that the trajectory is able to cross the linear piece L1 the system has a small amplitude limit
cycle whose amplitude increases with λ. In Fig. 18, this occurs for λ ∈ (0, 0.412). For larger values of
λ, the trajectory moves into regime R2 without crossing the linear piece L1. As for PWL1,2,1 systems,
the range of values of λ for which this occurs is very small.
In Fig. 19-A we present graphs of the turning curves vrev versus λ for the parameters in Fig. 18
and three different values of η > ηcr = 0.39. The red curves η = 0.6 correspond to Fig. 18 (since
η2 = 0.6). The initial values of the trajectory in R2 are (v1, w2,0) = (0.33, 0.0166) (top panel) and
(v1, w2,0) = (0.33, 0.01) (bottom panel). The values of vrev increase, first slowly and then abruptly;
small changes in λ cause large increases in vrev marking the fact that the corresponding vector field
not longer allows the trajectory to cross the v-nullcline and moves to the right towards the following
40
(v, w) η r1 r2 r µ set
Ll,1 (0, 0) -1.0 -0.052 -0.958 S - N I
Lr,1 -1.0 -0.052 -0.958 S - N I
L1 (0.33, 0.033) 0.1 0.192 0.045 U - F II
L2 (0.66, 0.231) 0.6 0.525 0.065 U - N II
L3 (1, 1) 2.262 2.244 0.008 U - N II
L1 (0.33, 0.033) 0.1 0.192 0.045 U - F III
L2 (0.66, 0.132) 0.3 0.126 0.145 U - F III
L3 (1, 1) 2.553 2.537 0.006 U - N III
L1 (0.33, 0.033) 0.1 0.192 0.045 U - F IV
L2 (0.66, 0.693) 2.0 1.980 0.010 U - N IV
L3 (1, 1) 0.903 0.857 0.036 U - N IV
Table 2: PWL1,3,1 models of FHN type for α = 4, ǫ = 0.01. The parameters correspond to three models
(sets I/II, I/III and I/IV) with the same left and right branches with a single linear piece each (set I).
For each linear piece Lj , the table shows the right endpoints (v, w), the slope η, the eigenvalues of the
corresponding linear regime (real eigenvalues r1 and r2 or real and imaginary parts, r = (η − ǫ)/2 and µ
respectively if the eigenvalues are complex). The transition from unstable foci (U-F) to unstable nodes (U-
N) occurs at η+cr = 0.39. The transition from stable foci (S-F) to stable nodes (S-N) occurs at η−cr = −0.41.
41
A B
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.0
Nv(v)
Nw
(v)
L1
L2
L3
Lr,1
Ll,1
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.412
Nv(v)
Nw
(v)
C D
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.432
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.433
Nv(v)
Nw
(v)
Figure 18: Canard explosion in a PWL1,3,1 system for α = 4 and ǫ = 0.01. A Nullclines for
λ = 0. B Small amplitude limit cycle where the trajectory crosses L1 almost at its joint point with the
linear piece L2. C Small amplitude limit cycle where the trajectory crosses L2. D Large amplitude limit
cycle. The canard explosion occurs for values of λ ∈ (0.432, 0.433). The canard critical value λc can be
well approximated by the value of λ corresponding to C. The corresponding parameters are given in Table
2 (sets I and II).
42
regime (R3). This is in contrast to what happens in the “focus” case where the turning curves depend
linearly with λ. The range of values of λ for which the trajectory is able to cross the v-nullcline
decreases with increasing values of η; i.e., the larger the slope of the linear piece L2 the shorter the
range of intervals of λ for which the intersection between the v-nullcline and the trajectory are possible.
As the initial value of the trajectory in R2 decreases, the turning point curves move to the left (see
bottom panel.); i.e., the trajectory’s ability to cross the linear piece L2 decreases. As w2,0 decrease
further the turning curves “disappear” to the left and trajectories can no longer cross the linear piece
L2.
In Fig. 19-B we present similar curves for α = 2 and w2,0 = 0.0166 (top panel) and w2,0 = 0.0136
(bottom panel). The behavior is similar to α = 4. By comparing the top panels of Fig. 19-B we
observe that as α decreases the ability of trajectories to cross the linear piece L2 also decreases. The
value of w2,0 used in the bottom panel in Fig. 19-B is larger than that used in Fig. 19-A.
Two additional representative examples are shown in Fig. 20. The corresponding parameters are
presented in Table 2 (sets III and IV). These are the same as in Fig. 18 with the exception of η2 and
η3 that has to satisfy the following constraint
η3 =1− η2 v21− v2
. (41)
In Fig. 20-A η2 is smaller than in Figs. 18 and R2 has an unstable focus rather than an unstable
node. Similarly to the behavior in R1, the canard explosion occurs when the vrev > v2 and the
trajectory is able to move to R3. In Fig. 20-B η2 = 2 is larger than in Fig. 18-B (η2 = 0.6) and R2
has an unstable node as in Fig. 18. However, differentely from Fig. 18, this value of η2 is too large
for the turning curves to exist so the trajectory is not able to cross the linear piece L2 and moves to
R3.
5 Discussion
Classical (smooth) models of FitzHugh-Nagumo (FHN) type (1) are prototypical caricature models of
excitable and oscillatory systems in a variety of fields including neuroscience, chemistry and biology
[8, 9, 10, 11]. When there exists a time-scale separation between the two participating variables
(ǫ ≪ 1), these models exhibit abrupt transitions between small and large amplitude oscillations (SAOs
and LAOs), known as canard explosions, as a control parameter (λ) moves across a very small critical
range [1, 2, 3, 4, 5, 6, 7]. In the context of neural models SAOs and LAOs correspond to subthreshold
membrane potential oscillations and action potentials respectively [39, 40, 41, 42].
In this paper, we have investigated the dynamics of two-dimensional, piecewise linear (PWL)
models of FHN type (2). PWL1,1,1 models of FHN type (Figs. 3-A and -B, top-left panels) display
oscillations in a single amplitude regime (around the left and right branches of the v-nullcline) and
hence they do not support canard-like transitions that require two amplitude modes of operation.
Canard-like transitions have been shown to occur, numerically, in PWL1,2,1 models of FHN type [33].
In this paper, using geometric and dynamic tools, we have provided a mechanistic explanation of the
dynamics of these models including the canard-like explosion of limit cycles, we have shown that the
43
A B
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 4
η = 0.4η = 0.5η = 0.6
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 2
η = 0.28η = 0.3η = 0.35
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 4
η = 0.4η = 0.5η = 0.6
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
λ
v rev
α = 2
η = 0.28η = 0.3η = 0.35
Figure 19: Reversing (or turning) points vrev in L2 as a function of λ for various values of α, η
and initial points (v0, w0) in L2. The parameters we used are: (v, w) = (0.33, 0.033), v0 = v, ǫ = 0.01,
A α = 4, ηcr = 0.39, η = 0.4 (r1 = 0.240 and r2 = 0.140) η = 0.5 (r1 = 0.040 and r2 = 0.087) η = 0.6
(r1 = 0.053 and r2 = 0.065) w0 = 0.0166 (top panel), w0 = 0.01 (bottom panel). B α = 2, ηcr = 0.273,
η = 0.28 (r1 = 0.167 and r2 = 0.103) η = 0.3 (r1 = 0.208 and r2 = 0.082) η = 0.35 (r1 = 0.281 and
r2 = 0.059) w0 = 0.0166 (top panel), w0 = 0.0136 (bottom panel).
44
A
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.537
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.538
Nv(v)
Nw
(v)
B
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.412
Nv(v)
Nw
(v)
−1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
v
w
α = 4 λ = 0.413
Nv(v)
Nw
(v)
Figure 20: Canard explosion in a PWL1,3,1 system for α = 4 and various piecewise linear
configurations in the middle branch of the v-nullcline. The corresponding parameters are given in
A-B: Table 2, set III and C-D: Table 2, set IV.
45
mechanistic principles extracted from this study can be extended to PWL1,3,1 models, and we argue
that they can be further extended to the investigation of PWL models with a larger number of linear
pieces. In addition, the tools we develop and discuss in this paper are amenable for the investigation of
a variety of related, and more complex, problems including forced, stochastic and coupled oscillators
of FHN type [42, 43, 44, 45, 38].
For a PWL1,2,1 model, a small amplitude limit cycle is created provided the limit cycle trajectory
is able to cross the middle branch of the v-nullcline which consists of the linear pieces L1 and L2.
Otherwise, a large amplitude limit cycle is created. If the linear regime R1 (corresponding to L1)
has an unstable node, then no small amplitude limit cycles exist; due to the difference in time-scales
between the participating variables, the limit cycle trajectory moves fast along an almost horizontal
direction and crosses the stable branch of the v-nullcline but never its middle branch. On the other
hand, if the linear regime R1 has an unstable focus, then the limit cycle trajectory is able to cross
the linear piece L1 and generate a small amplitude oscillation provided L1 is long enough for this
intersection to occur within R1. This picture is analogous to the dynamics generated in smooth
cubic-like systems undergoing a sub-critical and super-critical Hopf bifurcations respectively.
For a given size of the linear piece L1, we showed that the maximum value of the control parameter
λ for which the limit cycle trajectory crosses L1 can be approximated analytically from the model
parameters. In addition, we showed that this value of λ gives a good approximation to the numerically
computed canard critical value λc, and its dependence on the model parameters is consistent with the
canard critical value computed by other authors for smooth systems [2, 6]. Limit cycle trajectories
moving to the linear regime R2 can still cross the linear piece L2 and generate small amplitude limit
cycles for slightly larger values of λ but that range of values of λ for which this occurs is very small
(order of magnitude of ǫ).
The calculations mentioned above require knowledge of the initial conditions of the limit cycle
trajectory in R1 which in turn would require knowledge of this trajectory in the previous regimes
eventually going back to R1. However, time-scale separation allows for the estimation of these initial
conditions in R1 using asymptotic techniques thus making this estimation independent of the infor-
mation from previous regimes. Note, however, that for larger values of ǫ the time-scale separation
dilutes and this approximation becomes less accurate.
The considerations in the previous paragraph provide a heuristic algorithmic rule for deciding to
a good degree of accuracy whether a given set of parameters will give rise to either SAOs or LAOs.
Given the initial conditions of the limit cycle trajectory in R1, the slope η1 of the linear piece L1, and
the v-coordinate v1 of the joint point between the linear pieces L1 and L2:
• If R1 has an unstable node, then the system displays LAOs.
• If R1 has an unstable focus , then compute the v-coordinate vrev,1 of the intersection point
between the limit cycle trajectory and the linear piece L1.
– If vrev,1 ≤ v1, then the system displays SAOs.
– If vrev,1 > v1, then the system displays LAOs.
The last step in this algorithm is based on the fact that if R1 has a focus, then R2 has a node. The
frequency of the LAOs can be calculated following standard procedures for relaxation oscillators (see
46
[9]) for example.). The frequency of SAOs is given by the reciprocal of the sum of the times the small
amplitude limit cycle trajectory spends moving in a vicinity of the left branch of the v-nullcline and
on R1 (the transition from the latter to the former regimes is very fast). The frequency component
corresponding to the latter is given by µ in (12) with η substituted by η1.
The previous algorithm can be extended to PWL1,3,1 systems. Given the initial conditions of the
limit cycle trajectory in R1, the slopes η1 and η2, and the v-coordinates v1 and v2 (v-coordinates of
the joint points between the linear pieces L1 & L2 and L2 & L3 respectively):
• If R1 has a node (unstable), then the system displays LAOs.
• If R1 has a focus (unstable), then compute the v-coordinate vrev,1 of the intersection point
between the limit cycle trajectory and the linear piece L1.
– If vrev,1 ≤ v1, then the system displays SAOs.
– If vrev,1 > v1, then (the trajectory moves to R2)
∗ if R2 has a node (unstable), then the system displays LAOs.
∗ if R2 has a focus (unstable), then (1) compute the w coordinate of the trajectory
at v1 (see Appendix B) and use it as initial condition for the trajectory in R2, and
(2) compute the v-coordinate vrev,2 of the intersection point between the limit cycle
trajectory and the linear piece L2.
· if vrev,2 ≤ v2, then the system displays SAOs.
· if vrev,2 > v2, then the system displays LAOs.
Althouth we do not provide a rigourous argument on the accuracy of this algorithm, specially as
the values of ǫ increase and the time scale separation dilutes, it illustrates the dynamics of the limit
cycle trajectory in the vicinity of the unstable branch of the v-nullcline, and it helps in establishing
an organizing principle to understanding the dynamics of systems with a larger number of linear
pieces. As long as the limit cycle trajectory is in a “focus-regime”, it evolves by spiraling out around
the virtual fixed-point of the corresponding linear piece. It can either cross the v-nullcline, thus
generating a small amplitude limit cycle, or move to the next regime. Only when the trajectory
arrives to a “node-regime”, it escapes towards the right branch of the v-nullcline thus generating a
large amplitude limit cycle. The “last” value of λ for which the trajectory is able to cross a linear
piece corresponding to a “focus-regime” provides a good approximation to the canard critical value.
We conjecture that this is still true for systems with a larger number of linear pieces in the middle
branch. Note that a larger number of linear pieces in either of the other two branches (left and right)
has almost no effect on the dynamics of the system in the middle-branch regimes provided there is
a well defined time-scale separation between the two participating variables. The frequency profile
of the SAOs includes the frequencies µ corresponding to all the “foci-regimes” the trajectory visits,
possibly weighted by the size of the corresponding linear piece (or any other measure characterizing
the linear regimes).
A salient feature of the dynamics of fast-slow systems exhibiting the canard phenomenon is that
trajectories evolve in close vicinities of the unstable (middle) branch of the v-nullcline for a significant
47
amount of time before creating either SAOs or LAOs, as we illustrate in Fig. 1-A (middle and bottom
panels) for a prototypical smooth system of FHN type. Smooth cubic-like functions can be “piecewise-
linearized” following the procedure described in Section 2.2. For the example in Fig. 1-A, the PWL
middle branch consists of a sequence of linear pieces whose slopes first increase and then decrease
(see Fig. 3). From left to right, they correspond to a “focus-”, “node-” and again “focus-” regimes
respectively. Following our reasoning in the previous paragraphs, trajectories stay close to the unstable
branches of the v-nullclines for a significant amount of time because they are spiraling out around the
virtual fixed-points corresponding to the regimes they are subsequently visiting.
The previous arguments can be carried to the limit as the size of the linear pieces goes to zero as
we illustrate in Fig. 3. In this case, linear pieces are substituted by the tangent line to the cubic-like
function at each of its points.
Previous work on canard explosion of limit cycles has assumed time-scale separation between the
two participating variables (ǫ ≪ 1) [1, 2, 3, 4, 5, 6, 7]. Our results and tools are applicable to
systems with a more diluted time-scale separation with values of ǫ intermediate between these and
O(1). In addition, they are applicable to the investigation of the dynamics of forced and coupled
oscillators including sub- and super-threshold resonance problems and the construction of phase or
spike-time response curves and spike-time difference maps to investigate the synchronization properties
of coupled oscillators. In these cases typical perturbation terms are sinusoidal, pulsatile or synaptic-
like with various different rise- and decay-times. Limit cycle trajectories evolving in vicinities of the
stable branches of the v-nullcline may not feel the effect to these perturbations since they rapidly relax
to their original trajectory. However, trajectories evolving in the middle branch may feel significant
effects. For instance, a forcing term on the first equation in PWL1,2,1 system will cause a displacement
of the v-nullcline with respect to the w-nullcline. This in turn causes changes dynamic displacement
of the virtual fixed-point, and in the function (24). As a result, orbits corresponding to SAOs in the
unperturbed regime may transiently become LAOs and vice-versa causing the advance or delay of
spikes by a significant amount of time with respect to the unperturbed oscillation.
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Acknowledgments
This work was partially supported by the National Science Foundation grant DMS-0817241 (HGR).
SC would like to acknowledge discussions with Kyle Wedgwood regarding canards in PWL systems.
A Hopf bifurcation and canard critical values for two-
dimensional relaxation oscillators
The general form of a two-dimensional relaxation oscillator is given by
v′ = F (v,w),
w′ = ǫ G(v,w;λ).(42)
System (42) is fast-slow for ǫ ≪ 1. The zero level curves F (v,w) = 0 and G(v,w) = 0 define the
nullclines of system (42). We assume that the curve F (v,w) = 0 can be expressed as w = f(v)
with f(v) a cubic-like function having one local minimum at (vm, wm) and one local maximum at
(vM , wM ) with vm < vM and wm < wM . In the example shown in Fig. 1, f(v) = −2 v3 + 3 v2,
(vm, wm) = (0, 0) and (vM , wM ) = (1, 1). Without loss of generality, in the following discussion we
take (vm, wm) = (0, 0). The function G(v,w;λ) is assumed to be a non-increasing function of w
such that the zero level curve G(v,w;λ) = 0 is an increasing function of v for every λ in a given
neighborhood of λ = 0, and is also a decreasing function of λ for all v in a neighborhood of vm. In the
51
example shown in Fig. 1, G(v,w;λ) is expressed as w = g(v;λ) = αv − λ. We further assume that
F (v,w) = 0 and G(v,w;λ) intersect at (v, w) = (vm, wm) when λ = 0 and v > vm (v < vm) when
λ > 0 (λ < 0). The FitzHugh-Nagumo (FHN) and van der Pol (VDP) systems discussed below in
Sections A.3 and A.4 are particular cases of system (42).
A.1 Hopf bifurcation
For λ = 0, the nullclines of system (42) are assumed to intersect at the minimum of the cubic-like
nullcline F (v,w) = 0; i.e., (v, w) = (vm, wm). A Hopf bifurcation occurs when λ crosses the Hopf
bifuraction point λH > 0 given by ([5, 46])
λH(√ǫ) = − Gw
2 (Gv)1/2 |Fw|1/2ǫ+O(ǫ3/2), (43)
where all the functions are calculated at (vmin, wmin, λ = 0). For values of λ < λH , fixed points (v, w)
are stable. At the Hopf bifurcation the fixed point (v, w) becomes unstable and a small amplitude
limit cycle is created. This limit cycle is stable or unstable according to the Hopf bifurcation being
supercritical or subcritical respectively. The type of criticality of the Hopf bifurcation can be deter-
mined by looking at the sign of the criticality parameter ∆ which depends on the parameters of the
model and is given by [5, 46]
∆ =2
|Fw|1/2 (Gv)1/2 (Fvv)2Ω (44)
with
Ω = Gv Fvv Fvw + |Fw|Gv Fvvv − |Fw|Gvv Fvv −Gw (Fvv)2, (45)
where all the functions are calculated at (vmin, wmin, λ = 0). The Hopf bifurcation is supercritical
if ∆ < 0 (Ω < 0) and subcritical if ∆ > 0 (Ω > 0). If the Hopf bifurcation is subcritical (∆ > 0),
trajectories starting inside the unstable limit cycle converge to the stable fixed point (v, w). On the
other hand, if the Hopf bifurcation is supercritical (∆ < 0), trajectories starting inside the limit cycle
are attracted to it.
A.2 The canard phenomenon and the canard critical value
Here we follow Krupa and Szmolyan [5, 6]. As mentioned above, the canard explosion occurs in a
range of values of λ which is exponentially small in ǫ. For the purpose of our discussion we will refer
to this range as λc. The following approximation to the canard critical value λc has been calculated
as a function of the parameters of the model [5, 6, 43]
λc(√ǫ) = − Gv
2F 3vv |Gλ|
[Gv Fvw Fvv +Gv |Fw|Fvvv − |Fw|Gvv Fvv +Gw F 2vv ] ǫ+O(ǫ3/2), (46)
52
where all the functions are calculated at the point (vmin, wmin) and λ = 0 or, more generally, for the
value of λ corresponding to the intersection between the two nullclines ocurring at the minimum of the
cubic-like one. Eq. (46) has been calculated under the following assumptions [6] that define a canard
(fold) point and will be referred to as the canard conditions. They ensure that the nullclineG(v,w, λ) =
0 (w-nullcline) is transverse to the nullcline F (v,w) = 0 (v-nullcline), and it passes through the fold
point with non-zero speed as λ varies. Without loss of generality, we take (vm, wm) = (0, 0). (1)
F (0, 0) = 0, ∂F/∂v(0, 0)) = 0, ∂2F/∂v2(0, 0) 6= 0, i.e., (0, 0) is a nondegenerate local minimum (fold
point) of the nullcline F (v,w) = 0 for λ in a suitable interval. (2) ∂F/∂w(0, 0) 6= 0. (3) G(0, 0, 0) = 0,
∂G/∂v(0, 0, 0) 6= 0 and ∂G/∂λ(0, 0, 0) 6= 0. The Van der Pol and FitzHugh-Nagumo models discussed
below satisfy these conditions.
A.3 Van der Pol type models
Van der Pol (VDP) type models have the following general form
v′ = f(v)− w,
w′ = ǫ [v − λ],(47)
where f(v) is cubic-like. Here we assume, without lost of generality, that (vmin, wmin) = (0, 0). The w-
nullcline is a vertical line intersecting the v-axis at v = λ. The fixed point is given by (v, w) = (λ, f(λ)).
From eqs. (43), (44), (45) and (46), the Hopf bifurcation point, the Hopf bifurcation criticality
parameter, and the canard critical value are given respectively by
λH = O(ǫ3/2) ∼ 0 (> 0), ∆ =1
2 (f ′′)2f ′′′, λc = − 1
2 (f ′′)3f ′′′. (48)
If f(v) = −h v3 + a v2, then
∆ = −3h
a2< 0, λc =
3h
8 a3. (49)
Note that for VDP type oscillators the Hopf bifurcation (and the canard phenomenon) is always
supercritical.
A.4 FitzHugh-Nagumo type models
FitzHugh-Nagumo type models have the following general form
v′ = f(v)− w,
w′ = ǫ [ g(v;λ) − w ],(50)
where f(v) is cubic like and g(v;λ) is either linear of sigmoid-like. Here we assume, without lost of
generality, that (vmin, wmin) = (0, 0). From eqs. (43), (44), (45) and (46), the Hopf bifurcation point,
the Hopf bifurcation criticality parameter, and the canard critical value are given respectively by
λH =1
2 (g′)1/2ǫ+O(ǫ3/2), ∆ =
2
(g′)1/2 (f ′′)2[ (f ′′)2 + g′ f ′′′ − g′′ f ′′ ], (51)
53
and
λc =g′
2 (f ′′)3 |gλ|[(f ′′)2 − g′ f ′′′ + g′′ f ′′ ] ǫ+O(ǫ3/2). (52)
If f(v) = −h v3 + a v2, then
∆ =1
(g′)1/2 a2[ 2 a2 − 3h g′ − a g′′ ], λc =
g′
8 a3 |gλ|[ 2 a2 + 3h g′ + a g′′ ] ǫ+O(ǫ3/2). (53)
If, in addition, g(v;λ) = α v − λ, then
λH =1
2α1/2ǫ+O(ǫ3/2), ∆ =
1
α1/2 a2[ 2 a2 − 3α h ], λc =
α
8 a3[ 2 a2 +3αh ] ǫ+O(ǫ3/2). (54)
Note that the Hopf bifurcation is supercritical (subcritical) if α > 2a2/(3h) (α < 2a2/(3h)).
B Dynamics of the basic linear components
B.1 Calculation of target values for ( η + ǫ )2 − 4 ǫ α > 0 (real eigen-
values)
From (8) and (9), the time t it takes for the solution (V,W ) = (v− v, w− w) to evolve from the initial
conditions (V0,W0) to the target point (VT ,WT ) is given by
t =1
r2ln
(r2 + ǫ )VT −WT
(r2 + ǫ )V0 −W0. (55)
Given VT , WT must satisfy the following equation:
[V0 (r1 + ǫ ) −W0 ]
[
VT (r2 + ǫ ) −WT
V0 (r2 + ǫ ) −W0
]
r1r2 − [VT (r2 + ǫ ) −WT ] = VT (r1 − r2). (56)
Given WT , VT must satisfy the following equation:
(r2 + ǫ ) [V0 (r1 + ǫ ) −W0 ]
[
VT (r2 + ǫ ) −WT
V0 (r2 + ǫ ) −W0
]
r1r2 −
−(r1 + ǫ ) [VT (r2 + ǫ ) −WT ] = WT (r1 − r2). (57)
54
B.2 Calculation of the intersection point between trajectories and
a linear piece of the v-nullcline for ( η+ ǫ )2 − 4 ǫ α < 0 (complex eigen-
values)
Here we calculate the intersection points (vrev, wrev) between a trajectory given by eqs. (10) and (11),
initially located at (v0, w0), and a linear piece in the middle branch of the v-nullcline, described by
w = η (v − v) + w with η > 0. We assume that initially the trajectory is below this line; i.e., the
trajectory moves to the right and upwards. If the initial points of the trajectory and the linear piece
coincide (v0 = v) then this implies that w0 < w. We use the notation given in (11), (12) and (13).
At the intersection point (vrev, wrev), if it exists, the trajectory reverses direction (changing its
direction of motion from left-to-right to right-to-left). The time trev at which this happens satisfies
dv/dt = 0 with dw/dt 6= 0. By differentiating the first eq. in (10) we get
[ (c2 µ+ c1 r) cos(µ t) + (c2 r − c1 µ) sin(µ t) ] er t = 0. (58)
Thus, trev is the smallest positive number of the form:
trev =1
µtan−1 c2 µ+ c1 r
c1 µ− c2 r, (59)
and
vrev = v(trev), wrev = w(trev). (60)
Note that if trev is negative, then a multiple of π should be added to it. Note also that trev, and so
(vrev, wrev), depend on the difference between the initial and fixed points, (v0 − v, w0 − w) and not on
each of these points separately.
Linear piece L1
For the linear piece L1, v = w = 0 and v0 = 0. We further assume w0 = −λ ǫ. Then
v0 − v =−λ
α− η, w0 − w =
−λ (ǫ α− ǫ η + η)
α− η. (61)
Substituting into (9) and rearranging terms yields
c1 =−λ
α− η, c2 µ =
−λ
α− η
ǫ− η − 2 ǫ (α− η)
2. (62)
Substituting c1, c2, µ and r intro eq. (59) gives a simplified expression for trev
trev =1
µtan−1 −2µ
2 + η − ǫ. (63)
From eq. (63),
sin (µ trev) = − 2µ
2 + η − ǫcos (µ trev).
55
Substituting into the solution for v in (63) operating algebraically we obtain the following simplified
expression:
vrev = v(trev) =2λ
α− η
−1− η + ǫ− ǫ α+ ǫ η
2 + η − ǫcos (µtrev) e
(η−ǫ) trev/2. (64)
Clearly, vrev is a linear function of λ with slope
κ(η, ǫ, α) =2
α− η
−1− η + ǫ− ǫ α+ ǫ η
2 + η − ǫcos (µ trev) e
(η−ǫ) trev/2. (65)
B.3 Calculation of the intersection point between the trajectory and
a linear piece of the v-nullcline for ( η+ǫ σ)2−4 ǫ α > 0 (real eigenvalues)
Here we calculate the intersection points (vrev, wrev) between a trajectory given by eqs. (8) and
(9), initially at (v0, w0), and a linear piece in the middle branch of the v-nullcline, described by
w = η (v − v) + w with (η > 0). We assume that initially the trajectory is below this line; i.e., the
trajectory moves to the right and upwards. If the initial points of the trajectory and the linear piece
coincide (v0 = v) then this implies that w0 < w. We use the notation
r1,2 =η − ǫ σ ±
√
(η + ǫ σ)2 − 4 ǫ α
2, (66)
c1 =(v0 − v) (r1 + ǫ σ)− (w0 − w)
r1 − r2, c2 =
−(v0 − v) (r2 + ǫ σ) + (w0 − w)
r1 − r2. (67)
At the intersection point (vrev, wrev), if it exists, the trajectory reverses direction (changing its direction
of motion from left-to-right to right-to-left). The time trev at which this happens satisfies dv/dt = 0
with dw/dt 6= 0. By differentiating the first eq. in (8) we get
c1 r1 er1t + c2 r2 e
r2t = 0. (68)
Thus
trev =1
r1 − r2ln
−c2 r2c1 r1
, (69)
and
vrev = v(trev), wrev = w(trev). (70)
Note that r1 − r2 =√
(η + ǫ σ)2 − 4α ǫ.
56
B.4 Asymptotic behavior of trajectories approximating the limit
cycle in the vicinity of the stable branches of the PWL cubic-like
nullcline
We consider the FHN type system (1) expressed in terms of the slow variable τ = ǫ t (ǫ ≪ 1).
ǫ v = f(v)− w,
w = α v − λ− w ,(71)
in a domain where f ′(v) < 0. In (71) “dot” stands for d/dτ . For ǫ = 0, w = f(v). We follow [2]
and investigate the behavior of trajectories in O(ǫ) neighborhoods of w = f(v) by introducing the
transformation
w = f(v) + ǫ φ. (72)
Substitution into (71) yields
dv
dτ= −φ. (73)
By differentiating (72) with respect to τ , substituting (73) and using the second equation in (71)
we obtain
ǫdφ
dτ= f ′(v)φ+ α v − λ− f(v)− ǫ φ, (74)
or
dφ
dt= f ′(v)φ+ α v − λ− f(v)− ǫ φ. (75)
The steady state of eq. (74) is given by
φ =f(v) + λ− α v
f ′(v)− ǫ. (76)
To a first order approximation in ǫ (ǫ ≪ 1) trajectories move fast towards the slow manifold
approximated by w = f(v) + ǫ φ.
For a linear piece, f(v) = η v (with η < 0). Then
φ(v) =(η − α) v + λ
η − ǫ, (77)
limv→0
φ(v) =λ
η − ǫand lim
v→0w = ǫ φ = ǫ
λ
η − ǫ. (78)
57
B.5 Calculation of the inflection lines
For a general planar system of the form
v′ = F (v,w), w′ = G(v,w) (79)
a point of inflection is defined by
d
dv
(
dw
dv
)
=F dG
dv −GdFdv
F 2= 0. (80)
Hence, assuming it exists, an inflection point on a trajectory described by w = w(v) is determined by
the following geometric condition
F
[
∂G
∂v+
∂G
∂w
dw
dv
]
−G
[
∂F
∂v+
∂F
∂w
dw
dv
]
= 0. (81)
If system (79) is linear with
F (v,w) = A11v +A12w + b1, G(v,w) = A21v +A22w + b2, (82)
then eq. (81) defines two straight lines of the form w = w±(v), where
w±(v) =v[A21σ± −A11] + b2σ± − b1
A12 −A22σ±, (83)
σ± =A11 −A22 ±
√
(TrA)2 − 4 detA
2A21, (84)
and
A =
(
A11 A12
A21 A22
)
. (85)
The inflection lines w± only exist if the solutions of (83) are real, which is the case if the matrix
A has real eigen-values.
For a linear system of the form (4),
A =
[
ηj −1
ǫα −ǫ
]
and
[
b1b2
]
=
[
−ηj vj−1 + wj−1
−ǫλ
]
. (86)
Substituting into (83) and (84) we obtain the inflection line (25). Note that for the singular value
ǫ = 0 one of the inflection lines coincides with the voltage nullcline. We define the canard inflection
point as the set of parameters for which a periodic orbit develops a point of inflection.
58
B.6 Proof that the inflection and eigenvector lines coincide
Here we show that the inflection lines (25) and the eigenvector lines (26) coincide. The latter have the
eigenvector directions and cross the virtual fixed-point (v2, w2). In the remaining of this section we
will drop the subindex from the slope η, the fixed-point (v, w) and the endpoints (v, w). We proceed
by showing that both the slopes and the ordinates to the origin coincide; i.e.,
αη − r1,2 − ǫ
α− r1,2 − ǫ= r2,1 + ǫ (87)
and
λ (r1,2 + ǫ)− α (η v − w)
α− r1,2 − ǫ= −(r2,1 + ǫ) v + w. (88)
First we show that the identity (87) is true. Substituting (6) into (87) gives
α2 η − (η − ǫ±
√P )− 2 ǫ
2α − (η − ǫ±√P )− 2 ǫ
= r2,1 + ǫ (89)
where
P = (η + ǫ)2 − 4 ǫ α (90)
Substituting (90) into (89) and rearranging terms gives
αη − ǫ∓
√P
2α− η − ǫ∓√P
= r2,1 + ǫ (91)
or
αr2,1
α− η + r2,1= r2,1 + ǫ. (92)
Multiplying the denominator in the left hand side of eq. (92) by its right hand side, rearranging terms
and using the fact that r2,1 are roots of the characteristic polynomial of the matrix A in (86) gives
the numerator in the left hand side in eq. (92) showing that the identity (87) is true.
Now we show that the identity (88) is true. We will use the following identity
r2,1 + ǫ = η − r1.2, (93)
that can be calculated using (90), and
η v − w = λ− (α− η) v, (94)
that can be calculated from (5). Substituting (93) and (94) into (88) and rearranging terms we obtain
(λ+ w ) ( r1,2 + ǫ− α ) = [ (r1,2 − η) (α− ǫ− r1,2)− α (α− η) ] v. (95)
59
Manipulating algebraically the right hand side of eq. (95), using the fact that r2,1 are roots of the
characteristic polynomial of the matrix A in (86) and rearranging terms we obtain
(λ+ w ) ( r1,2 + ǫ− α ) = α ( r1,2 + ǫ− α ) v (96)
or
(α v − λ− w ) ( r1,2 + ǫ− α ) = 0 (97)
which follows from the fact that (v, w) is a fixed-point of the linear system and w = α v − λ is its
w-nullcline.
60
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